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OF  CALIFORNIA 

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TREATISE 


ON 


LAND-SURVEYING: 

COMPRISING 

THE  THEORY 

DEVEIOPED  FROM  FIVE  ELEIIIENTARY  PROCIPLES ; 

AND  THE  PEAGTICE 


WITH  THE  CHAIN  ALONE,  THE  COMPASS,  THE  TRANSIT, 
THE  THEODOLITE,  THE  PLANE  TABLE,  &c. 


ILLUSTRATED    BT 

FOUR  HUNDRED  ENGRAVINGS, 

AND  A  MAGNETIC  CHART. 


By  W.  M.  GILLESPIE,  LL.D.,  Civ.  Eitg., 

PKOFESSOR   OF  CIVIL  ENGINEERING  IN  UlS'ION  COLLEGE. 
AUTHOB  OF    "  A  MANUAL  OF  BOAD- MAKING,"  ETC. 


EIGHTH    EDITION. 

NEW     YORK: 

D.  APPLETON  &  COMPANY,  549  &  551   BROADWAY. 

LONDON  :    16  LITTLE  BKITAIN. 

187.5. 

JOH|V  S.  PRELL 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 


Eutered,  according  to  the  Act  of  Congress,  m  the  year  1855,  by 

WILLIAM  MITCHELL  GILLESPIE, 

III  tie  (Jlevk's  Office  of  the  District  Court  of  the  United  States  for  the  SoutherE 
District  of  New- York. 


Rn^ineeriBg 
Library 


TA 

JOHIM  S.  PRELL  ^^'^ 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 

PREFACE. 


Land-Surveying  is  perhaps  the  oldest  of  the  mathematical  arts.  Indeed, 
Geometry  itself,  as  its  name — "Land-measuring" — implies,  is  said  to 
have  arisen  from  the  eflforts  of  the  Egyptian  sages  to  recover  and  to  fix 
the  land-marks  annually  swept  away  by  the  inundations  of  the  Nile. 
The  art  is  also  one  of  the  most  important  at  the  present  day,  as  deter- 
mining the  title  to  land,  the  foundation  of  the  whole  wealth  of  the 
world.  It  is  besides  one  of  the  most  useful  as  a  study,  from  its 
striking  exemplifications  of  the  practical  bearings  of  abstract  mathematics. 
But,  strangely  enough,  Surveying  has  never  yet  been  reduced  to  a  system- 
atic and  symmetric  whole.  To  effect  this,  by  basing  the  art  on  a  few 
simple  principles,  and  tracing  them  out  into  their  complicated  ramifications 
and  varied  applications  (^ which  extend  from  the  measurement  of  "a  mow- 
ing lot  "  to  that  of  the  Heavens) ,  has  been  the  earnest  endeavor  of  the- 
present  writer. 

The  work,  in  its  inception,  grew  out  of  the  author's  own  needs.  Teach- 
mg  Surveying,  as  preliminary  to  a  course  of  Civil  Engineering,  he 
found  none  of  the  books  in  use  (though  very  excellent  in  many  respects) 
suited  to  his  purpose.  He  was  therefore  compelled  to  teach  the  subject 
by  a  combination  of  familiar  lectures  on  its  principles  and  exemplifica- 
tions of  its  practice.  His  notes  continually  swelling  in  bulk,  gradually 
became  systematized  in  nearly  their  present  form,  and  in  1851  he  printed 
a  synopsis  of  them  for  the  use  of  his  classes.  His  system  has  thufi 
been  fully  tested,  and  the  present  volume  is  the  result. 

712634 


<T  LAIVD-SERVEYING. 

A  double  object  has  been  kept  in  view  in  its  preparation;  viz.  to 
produce  a  very  plain  introduction  to  the  subject,  easy  to  be  mastered  by 
the  young  scholar  or  the  practical  man  of  little  previous  acquirement, 
the  only  pre-requisites  being  arithmetic  and  a  little  geometry ;  and  at  the 
same  time  to  make  the  instruction  of  such  a  character  as  to  lay  a  founda 
tion  broad  enough  and  deep  enough  for  the  most  complete  superstructure 
which  the  professional  student  may  subsequently  wish  to  raise  upon  it 

For  the  convenience  of  tnose  wishing  to  make  a  hasty  examination  of 
the  book,  a  summary  of  some  of  its  leading  points  and  most  peculiar 
features  will  here  be  given.  * 

I.  All  the  operations  of  Surveying  are  deduced  from  only  five  simple 
principles.  These  principles  are  enunciated  and  illustrated  in  Chapter  1, 
of  Part  I.  They  will  be  at  once  recognized  by  the  Geometer  as  familiar 
systems  of  "  Co-ordinates ;"  but  they  were  not  here  arbitrarily  assumed  in 
advance.  They  were  arrived  at  most  practically  by  analyzing  all  the 
numerous  and  incongruous  methods  and  contrivances  employed  in  Sur- 
veying, and  rejecting,  one  after  another,  all  extraneous  and  non-essential 
portions,  thus  reducing  down  the  operations,  one  by  one  and  step  by 
step,  to  more  and  more  general  and  comprehensive  laws,  till  at  last, 
by  continual  elimination,  they  were  unexpectedly  resolved  into  these 
few  and  simple  principles ;  upon  which  it  is  here  attempted  to  build  up 
a  symmetrical  system. 

II.  The  three  operations  common  to  all  kinds  of  Land-surveying,  viz. 
Making  the  Measurements,  Drawing  the  Maps,  and  Calculating  the 
Contents,  are  fully  examined  in  advance,  in  Part  I,  Chapters  2,  3,  4^ 
so  that  when  the  various  methods  of  Surveying  are  subsequently  taken 
up,  only  the  few  new  points  which  are  peculiar  to  each,  require  to  be 
explained. 

Each  kind  of  Surveying,  founded  on  one  of  the  five  fundamental  prin- 
ciples, is  then  explained  in  its  turn,  in  the  successive  I'arts,  and  each 
carefully  kept  distinct  from  the  rest. 


Preface.  ▼ 

m.  A  complete  system  of  Surveying  with  only  a  chain,  a  rope,  o? 
any  substitute,  (invaluable  to  farmers  having  no  other  instruments,)  is 
very  fully  developed  in  Part  II 

IV.  The  various  Problems  in  Chapter  5,  of  Part  II,  will  be  founa 
to  constitute  a  course  of  practical  Geometry  on  the  ground.  As  some 
of  their  demonstrations  involve  the  "Theory  of  Transversals,  etc,"  (a 
beautiful  supplement  to  the  ordinary  Geometry),  a  carefully  digested 
summary  of  its  principal  Theorems  is  here  given,  for  the  first  time  in 
English.     It  will  be  found  in  Appendix  B. 

V.  In  Compass  Surveying,  Part  III,  the  Field  work,  in  Chapter  3, 
13  adapted  to  our  American  practice ;  some  new  modes  of  platting  bear- 
ings are  given  in  Chapter  4,  and  in  Chapter  6,  the  rectangular  method 
of  calculating  contents  is  much  simplified. 

VI.  The  effects  of  the  continual  change  in  the  Variation  of  the  mag 
netic  needle  upon  the  surveys  of  old  lines,  the  difficulties  caused  by  it, 
and  the  means  of  remedying  them,  are  treated  of  with  great  minuteness 
of  practical  detail.  A  new  table  has  been  calculated  for  the  time  of 
"greatest  Azimuth,"  those  in  common  use  being  the  same  as  the  one 
prepared  by  Gummere  in  1814,  and  consequently  greatly  in  error  now 
from  the  change  of  place  of  the  North  Star  since  that  date. 

VII.  In  Part  IV,  in  Chapter  1,  the  Transit  and  Theodolite  are 
explained  in  every  pointj;  in  Chapter  2,  all  forms  of  Verniers  are  shewn 
by  numerous  engravings;  and  in  Chapter  3,  the  Adjustments  are 
elucidated  by  some  novel  modes  of  illustration. 

VIII.  In  Part  VII,  will  be  found  all  the  best  methods  of  overcominc 

o 

obstacles  to  sight  and  to  measurement  in  angular  Surveying. 

IX.  Part  XI  contains  a  very  complete  and  systematic  collection  of 
the  principal  problems  in  the  Division  of  Land. 

X.  The  Methods  of  Surveying  the  Public  Lands  of  the  United  States, 
of  marking  lines  and  corners,  &c.,  are  given  in  Part  XII,  from  official 
documents,  with  great  minuteness ;  since  the  subject  interests  so  many 
land-owners  residing  in  the  Eastern  as  well  as  in  the  Western  States. 


n  LAND-SIRVEYING. 

XI.  The  'Tables  comprise  a  Traverse  Table,  computed  for  this  volume, 
nd  giving  increased  accuracy  in  one-fifteenth  of  the  usual  space;  a 
Tahle  of  Chords,  appearing  for  the  first  time  in  English,  and  supplying 
he  most  accurate  method  of  platting  angles;  and  a  Table  of  natural 

Sines  and  Tangents.  The  usual  Logarithmic  Tables  are  also  giveii. 
The  tables  are  printed  on  tinted  paper,  on  the  eye-saving  principle  of 
Babbage. 

XII.  The  great  number  of  engraved  illustrations,  most  of  them  crig- 
inal,  is  a  peculiar  feature  of  this  volume,  suggested  by  the  experience  of 
the  author  that  one  diagram  is  worth  a  page  of  print  in  giving  clearness 
and  definiteness  to  the  otherwise  vague  conceptions  of  a  student. 

XIII.  The  practical  details,  and  hints  to  the  young  Surveyor,  have 
been  made  exceedingly  full  by  a  thorough  examination  of  more  than  fifty 
works  on  the  subject,  by  English,  French  and  German  writers,  so  as  to 
make  it  certain  that  nothing  which  could  be  useful  had  been  overlooked. 
It  would  be  impossible  to  credit  each  item  (though  this  has  been  most 
Bcrupulously  done  in  the  few  cases  in  which  an  American  writer  has  been 
referred  to),  but  the  principal  names  are  these:  Adams,  Ainslie,  Baker, 
Begat,  Belcher,  Bourgeois,  Bourns,  Brees,  Bruff,  Burr,  Castle,  Fran- 
coeur,  Frome,  Galbraith,  Gibson,  Guy,  Hogard,  Jackson,  Lamotte, 
Lefevre,  Mascheroni,  Narrien,  Nesbitt,  Pearson,  Puille,  Puissant,  Beg- 
nault,  Richard,  Serret,  Simms,  Stevenson,  Weisbach,  Williams. 

Should  any  important  error,  either  of  printer  or  author,  be  discovered 
(as  is  very  possible  in  a  work  of  so  much  detail,  despite  the  great  care 
used)  the  writer  would  be  much  obliged  by  its  prompt  communication. 

The  present  volume  will  be  followed  by  another  on  Levelling  and 
Higher  Surveying  :  embracing  Levelling  (with  Spirit-Level,  Theodo- 
lite, Barometer,  etc.)  ;  its  applications  in  Topography  or  Hill-drawing, 
;n  Mining  Surveys,  etc. ;  the  Sextant,  and  other  reflecting  instruments  ; 
Maritime  Surveying ;  and  Geodesy,  with  its  practical  Astronomy. 


GENERAL  DIVISION  OF  THE  SUBJECT. 

[_A  full  Analytical  Tabic  of  Contents  is  given  at  the  end  of  the  vnlume.'\ 

PART  I.     GENERAL  PRINCIPLES  AND  OPERATIONS. 

Chapt.  1.  Definitions  and  Methods S 

Chapt.  2.  Making  the  Measurements 16 

Chapt.  3.  Drawing  the  Map 25 

Chapt.  4.  Calculating  th  e  Content 38 

PART  II.    CHAIN   SURVEYING. 

Chapt.  1.  Surveying  by  Diagonals 57 

Chapt.  2.  Surveying  by  Tie-Lines 66 

Chapt.  3.  Surveying  by  Perpendiculars 68 

Chapt.  4.  Surveying  by  these  Methods  combined 82 

Chapt.  5.  Obstacles  TO  Measurement 96 

PART  III.     COMPASS  SURVEYING. 

Chapt.  1.  Angular  Surveying  in  General 122 

Chapt.  2.  The  Compass 127 

Chapt.  3.  The  Field-Work 1.S8 

Chapt.  4.  Platting  the  Survey 157 

Chapt.  5.  Latitudes  and  Departures 169 

Chapt.  6.  Calculating  the  Content 180 

Chapt.  7.  3Iagnetic  Variation 189 

Chapt.  8.  Changes  in  the  Variation 208 

PART  IV.    TRANSIT  AND  THEODOLITE  SURVEYING. 

Chapt.  1.  The  Instruments 211 

Chapt.  2.  Verniers 228 

Chapt.  3.  Adjustments 240 

Chapt.  4.  The  Field-Work 250 

PART  V.    TllIANGULAR  SIRVEYING 260 

PART  VI.     TRILINEAR  SURVEYING 275 

PART  VII.     OBSTACLES  IN  ANGULAR  SURVEYING. 

Chapt.  1.   Perpendiculars  and  Parallels 279 

Chapt.  2.  Obstacles  to  Alinement 282 

Chapt.  3.  Obstacles  to  Measurement 287 

Chapt.  4.  Supplying  Omissions 297 

PART  VIII.     PLANE-TABLE  SURVEYING 303 

PART  IX.     SURVEYING  WITHOUT  INSTRUMENTS 311 

PART  X.     3IAPPING. 

Chapt.  1.  Copying  Plats 316 

Chapt.  2.  Conventional  Signs : 322 

Chapt.  3.  Finishing  the  Map 328 

PART  XI.     LAYING  OUT  AND  DIVIDING  UP  LAND. 

Chapt.  1.  Laying  out  Land 830 

Chapt.  2.  Parting  off  Land , 334 

Chapt.  3.  Dividing  up  Land 347 

PART  XII.     UNITED  STATES'  PUBLIC  LANDS 36? 

APPENDIX  A — Synopsis  of  Plane  Trigonometry 379 

APPENDIX  B — Demonstrations  of  Problems,  &c 387 

APPENDIX  C — iNTRODacTioN  to  Levelling 409 

ANALYTICAIj  table  of  CONTENTS 415 


TO  TEACHERS  AND  STUDENTS. 


As  it  is  desirable  to  obtain,  at  the  earliest  possible  period,  a  sufficient  knowledge  of  the  generm 
principles  of  Surveying  to  commence  its  practice,  the  Student  at  his  first  reading  may  omit  th» 
portions  indicated  below,  and  take  them  up  subsequently  in  connection  with  his  review  of  his  studie*. 
The  same  omissions  may  be  made  by  Teachers  whose  classes  have  only  a  short  time  for  this  ttudy. 

In  PART  I,  omit  only  Articles  (46),  (47),  (48),  (51),  (72),  (34),  (85). 

In  PART  II,  omit,  in  Chapter  IV,  (127),  (128),  (189),  (130);  and  in  Cliapter  V,  learn  at  Jim 
tuuter  each  Problem,  only  one  or  two  of  the  simpler  methods 

In  PART  III,  omit  only  (225),  (226),  232),  (233),  (244),  (251),  (280),  (297),  (322) 

Then  pass  over  PART  IV;  and  in  PART  V,  take  only  (379),  (380) ;  and  (391)  to  (39in 

Then  pass  over  PART  VI;  and  go  to  PART  VII,  (Jf  the  student  has  studied  Trigonometry)^ 
tnd  omit  (423) ;  (431)  to  (438) ;  and  all  of  Chapter  IV,  except  (439)  and  (44O). 

PART  VIII  may  be  passed  over;  and  PARTS  IX  and  X  may  be  taken  in  full. 

in  PART  XI,  take  all  of  Cluipter  I;  and  in  Chapters  II  and  III,  take  only  the  simpler  ccn 
ttruetions,  not  omittvig,  however,  (517),  (518)  and  (538). 

In  PART  Xn,  take  (560),  (561),  (565),  (566). 

Appendix  C,  on  LEVELLING,  may  conclude  this  dbridgra  sotirat. 


LAND-SURVEYIIG 


PAET  I. 

GENERAL   PRINCIPLES 

AND 

FUNDAMENTAL    OPERATIONS. 

CHAPTER  I. 

DEFINITIOIVS  AND  METHODS. 

(1)  Surveying  is  the  art  of  makiiig  such  measurements  as  will 
determine  the  relative  positions  of  any  points  on  the  surface  of  the 
earth ;  so  that  a  3Iap  of  any  portion  of  that  surface  may  be  drawn, 
and  its  Content  calculated. 

(2)  The  position  of  a  point  is  said  to  be  determined,  when  it  is 
known  how  far  that  point  is  from  one  or  more  given  poiuts,  and  in 
what  direction  there-from ;  or  how  far  it  is  in  front  of  them  or 
behind  them,  and  how  far  to  their  right  or  to  their  left,  &c ;  so 
that  the  place  of  the  first  pohit,  if  lost,  could  be  again  found  by 
repeating  these  measurements  in  the  contrary  direction. 

The  "  points"  which  are  to  be  determined  in  Surveying,  are  not 
the  mathematical  points  treated  of  in  Geometry;  but  the  comers 
of  fences,  boundary  stones,  trees,  and  the  like,  which  are  mere 
points  in  comparison  with  the  extensive  surfaces  and  areas  which 
they  are  the  means  of  determining.  In  strictness,  their  centres 
should  be  regarded  as  the  points  alluded  to. 


10  GENERAL  PRIIVCIPLES.  [part  i 

(3)  A  straight  Line  is  "  determined,"  that  is,  has  its  length 
and  its  position  known  and  fixed,  when  the  points  at  its  extrem- 
ities are  determined ;  and  a  plane  Surface  has  its  form  and  dimen- 
sions determined,  when  the  lines  which  bound  it  are  determined. 
Consequently,  the  determination  of  the  relative  positions  of  points 
is  all  that  is  necessary  for  the  principal  objects  of  Surveying ; 
which  are  to  make  a  map  of  any  surface,  such  as  a  field,  farm, 
state,  &c.,  and  to  calculate  its  content  in  square  feet,  acres,  or 
square  miles.  The  former  is  an  apphcation  of  Drafting,  the  latter 
of  Mensuration. 

(4)  The  position  of  a  point  may  be  determined  by  a  variety  of 
methods.  Those  most  frequently  employed  in  Surveying,  are  the 
following ;  all  the  points  being  supposed  to  be  in  the  same  plane. 

(5)  First  Method.  By  measuring  the  distances  from  the  re- 
quired  point  to  two  given  points. 

Thus,  m  Fig.  1,  the  pomt  S  is  "  deter-  ^'^-  ^-    ^ 

mined,"  if  it  is  known  to  be  one  inch  ^^^  \ 

from  A,  and  half  an  mch  from  B  :  for,  ^^'^  \ 

its  place,  if  lost,  could  be  found  by  de-      ji^.:^ \j^ 

scribing  two  arcs  of  circles,  from  A  and  B  as  centres,  and  with  the 
^ven  distances  as  radii.  The  required  point  would  be  at  the 
intersection  of  these  arcs. 

In  applying  this  prraciple  in  surveying,  S  may  represent  any 
station,  such  as  a  corner  of  a  field,  an  angle  of  a  fence,  a  tree,  a 
house,  &c.  If  then  one  corner  of  a  field  be  100  feet  from  a 
second  corner,  and  50  feet  from  a  third,  the  place  of  the  first  cor- 
ner is  known  and  determined  with  reference  to  the  other  two. 

There  will  be  two  points  fulfilling  this  condition,  one  on  each  side 
of  the  given  fine,  but  it  will  always  be  known  which  of  them  is  the 
one  desired. 

In  Creographi/,  this  principle  is  employed  to  indicate  the  posi- 
tion of  a  town ;  as  when  we  say  that  Buffalo  is  distant  (in  a  straight 
'  line)  295  miles  from  New-York,  and  390  from  Cincmnati,  and 
thus  convey  to  a  stranger  acquainted  with  only  the  last  two  places 
,  a  correct  idea  of  the  position  of  the  first. 


cuAPi]  Definitions  and  Metliods.  11 

In  Analytical  Geometry^  the  lines  AS  and  BS  are  knovra  aa 
^ Focal  Co-ordinates f^  the  general  name  "co-ordinates"  being 
applied  to  the  lines  or  angles  which  determine  the  position  of  a 
point. 

(6)  Second  method.  By  measuring  the  perpendicular  dis- 
tance  from  the  required  point  to  a  given  ■  line,  and  the  distance 
thence  along  the  line  to  a  given  point. 

Thus,  in  Fig.  2,  if  the  perpendicular  dis-  ,     Fig.  2. 

tance  SC  be  half  an  inch,  and  CA  be  one 
inch,  the  point  S  is  "  determined "  :  for,  its 
place  could  be  again  found  by  measui-ing  one 

inch  from  A  to  C,  and  half  an  inch  from  C,    S. c! 

at  right  angles  to  AC,  which  would  fix  the  point  S. 

The  Pubhc  Lands  of  the  United  States  are  laid  out  by  this 
method,  as  will  be  explained  in  Part  XII. 

In  Creography,  this  principle  is  employed  under  the  name  of 
Latitude  and  Longitude. 

Thus,  Philadelphia  is  one  degree  and  fifty-two  minutes  of  longi- 
tude east  of  Washington,  and  one  degree  and  three  minutes  of  lati- 
tude north  of  it. 

In  Analytical  Geometry,  the  lines  AC  and  CS  are  kno^-n  as 
'^Rectangular  Co-ordinates.''^  The  point  is  there  regarded  as 
determined  by  the  uitersection  of  two  hues,  drawn  parallel  to  two 
fixed  lines,  or  ".Aa^es,"  and  at  a  given  distance  from  them.  These 
Axes,  in  the  present  figure,  would  be  the  fine  AC,  and  another 
line,perpendicular  to  it  and  passing  through  A,  as  the  origin. 

(7)  Third  Method.  By  measuring  the  angle  hetiveen  a  given 
line  and  a  line  drawn  from  any  given  point  of  it  to  the  required 
point ;  and  also  the  length  of  this  latter  line. 

Thus,  in  Fig.  3,  if  we  know  the  angle  Ficr.  3. 

BAS  to  be  a  third  of  a  right  angle,  and  Sc 

AS  to  be  one  inch,  the  point  S  is  determin-  ^' 

ed  ;  for,  its  place  could  be  found  by  drawing  ^' 

from  A,  a  line  making  the  given  angle  with  A"^^- B 

AB,  and  measurmg  on  it  the  given  distance. 


12  GENERAL  PRINCIPLES.  [part  i 

In  applying  this  principle  in  surveying,  S,  as  before,  may  repre- 
Bent  any  station,  and  the  line  AB  may  be  a  fence,  or  any  other 
real  or  imaginary  line. 

In  "  Compass  Surveying,"  it  is  a  north  and  south  line,  the  direc- 
tion of  which  is  given  by  the  magnetic  needle  of  the  conipass. 

In  Creography,  this  principle  is  employed  to  determine  the  rela- 
tive positions  of  places,  by  "  Bearings  and  distances  " ;  as  when  we 
Bay  that  San  Francisco  is  1750  miles  nearly  due  west  from  St.  Louis ; 
the  word  "  west"  indicating  the  direction,  or  angle  which  the  line 
joining  the  two  places  makes  with  a  north  and  south  line,  and 
the  number  of  miles  giving  the  length  of  that  line. 

In  Analytical  Creometry,  the  line  AS,  and  the  angle  BAS,  are 
called  ^^  Polar  Oo-ordinates.^^ 

(8)  Fourth  method.  By  measuring  the  angles  made  with  a 
given  line  hy  two  other  lines  starting  from  given  j^oints  upon  it, 
and  passing  through  the  requh'ed  point. 

Thus,  in  Fig.  4,  the  point  S  is  deter-  Fig  4. 

mined  by  being  in  the  intersection  of  the  ^^ 

two  lines  AS  and  BS,  which  make  re-  ^^"^       \ 

spectively  angles  of  a  half  and  of  a  third      ^^'^' 
of  a  right  angle  with  the  line  AB,  which    a  .-b 

is  one  inch  long ;  for,  the  place  of  the  point  could  be  found,  if  lost, 
by  drawing  from  A  and  B  lines  making  with  AB  the  known  angles. 

In  G-eography,  we  might  thus  fix  the  position  of  St.  Louis,  by 
saying  it  lay  nearly  due  north  from  New-Orleans,  and  due  west 
from  Washington. 

In  Analytical  Creometry,  these  two  angles  would  be  called 
'"''Angular  Co-ordinates.^^ 

(9)  In  Fig.  6,  are  shown  together  all 
the  measurements  necessary  for  determin- 
mg  the  same  point  S,  by  each  of  the  four 
preceding  methods.  In  the  First  Me- 
thod, we  measure  the  distances  AS  and  ^ 
BS ;  in  the  Second  Method,  the  distances  AC  and  CS,  the  latter 
at  right  angles  to  the  former ;  in  the  Third  Method,  the  distance 


CHAP.  I.J  Definitions  and  Methods*  13 

AS,  and  the  angle  SAB ;  and  in  the  Fourth  Method,  the  angles 
SAB  and  SBA.  In  all  these  methods  the  point  is  really  deter- 
mined by  the  intersection  of  two  lines,  either  straight  lines  or 
arcs  of  circles.  Thus,  in  the  First  Method,  it  is  determined  by 
the  intersection  of  two  circles  ;  in  the  Second,  by  the  intersection 
of  two  straight  lines  ;  in  the  Third,  by  the  intersection  of  a  straight 
line  and  a  circle ;  and  in  the  Fourth,  by  the  intersection  of  two 
straight  lines. 

(10)  Fiftli  Method.  By  measuring  the  angles  made  with  each 
other  hy  three  lines  of  sight  passing  from  the  required  'point  to 
three  points  whose  positions  are  hioivn. 

Thus,  in  Fig.  6,  the  point  S  is  deter- 
mined by  the  angles,  ASB  and  BSC, 
made  by  the  three  lines  SA,  SB  and 
SC. 

Geographically,  the  position  of  Chi- 
cago would  be  determined  by  three 
straight  lines  passing  from  it  to  Wash- 
ington, Cincinnati,  and  Mobile,  and  mak- 
ing known  angles  with  each  other ;  that  of  the  first  and  second 
lines  being  about  one-third,  and  that  of  the  second  and  third  lines, 
about  one-half  of  a  right  angle. 

From  the  three  lines  employed,  this  may  be  named  the  Method 
of  Trilinear  co-ordinates. 

(11)  The  position  of  a  point  is  sometimes  determined  by  the 
intersection  of  two  lines,  which  are  themselves  determined  by  their 
extremities  bemg  given.  Thus,  in  Fig.  7,  Fiff-  7. 
the  point  S  is  detennined  by  its  being  sit- 
uated in  the  intersection  of  AB  and  CD. 
This  method  is  sometimes  employed  to  fix 
the  position  of  a  Station  on  a  RaU-Boad 
line,  &c.,  when  it  occurs  in  a  place  where 
a  stake  cannot  be  driven,  such  as  in  a  pond  \  and  in  a  few  other 
cases  ;  but  is  not  used  frequently  enough  to  require  that  it  should 
be  called  a  sixth  principle  of  Surveying. 


14  GENERAL  PRINCIPLES.  [part  i 

(12)  These  five  methods  of  determining  the  positions  of  points, 
produce  five  corresponding  systems  of  Surveying,  which  may  be 
named  as  follows : 


I.  DIAGONAL  SURVEYING. 
II.  PERPENDICULAR  SURVEYING. 

III.  POLAR  SURVEYING 

IV.  TRIANGULAR  SURVEYING. 
V.  TRILINEAR  SURVEYING. 

(13)  The  above  division  of  Surveying  has  been  made  in  har- 
mony with  the  principles  involved  and  the  methods  employed. 

The  subject  is,  however,  sometimes  divided  with  reference  to  the 
instruments  emphjedi;  as  the  chain,  either  alone  or  with  cross- 
stafi";  the  compass ;  the  transit  or  theodolite ;  the  sextant ;  the 
plane  table,  &c. 

(14)  Surveying  may  also  be  divided  according  to  its  objects. 

In  Land  Surveying,  the  content,  in  acres,  &c.,  of  the  tract  sur- 
veyed, is  usually  the  principal  object  of  the  survey.  A  map, 
showing  the  shape  of  the  property,  may  also  be  required.  Certain 
signs  on  it  may  indicate  the  different  kinds  of  culture,  &c.  This 
land  may  also  be  required  to  be  divided  up  in  certain  proportions ; 
and  the  lines  of  division  may  also  be  required  to  be  set  out  on  the 
ground.  One  or  all  of  these  objects  may  be  demanded  in  Land 
Surveying. 

In  Topographical  Surveying,  the  measurement  and  graphical 
representation  of  the  inequahties  of  the  ground,  or  its  "  relief,"  i.  e. 
its  hills  and  hollows,  as  determined  by  the  art  of  "  Levelling,"  ia 
the  leading  object. 

In  Maritime '  or  Sydro graphical  Surveying,  the  positions  of 
rocks,  shoals  and  channels  are  the  chief  subjects  of  examination. 

In  Mining  Surveying,  the  directions  and  dimensions  of  the  sub- 
terranean passages  of  mines  are  to  be  determined. 


CHAP,  i]  Definitions  and  Methods.  15 

(15)  Surveying  may  also  be  divided  according  to  the  extent  of 
the  district  surveyed,  into  Plane  and  Greodesio.  Geodesy  takes 
Into  account  the  curvature  of  the  earth,  and  employs  Spherical 
Trigonometry.  Plane  Surveying  disregards  this  curvature,  as  a 
needless  refinement  except  in  very  extensive  surveys,  such  as  those 
of  a  State,  and  considers  the  surface  of  the  earth  aa  plane,  which 
may  safely  be  done  in  surveys  of  moderate  extent. 

(16)  Land  Surveying  IS  the  principal  subject  of  this  volume;: 
the  surface  surveyed  being  regarded  as  plane ;  and  each  of  the 
five  Methods  being  in  turn  employed.  For  the  purposes  of  instruc- 
tion, the  subject  will  be  best  divided,  partly  with  reference  to  the 
Methods  employed,  and  partly  to  the  Instruments  used.  Accord- 
ingly, the  Pir^t  and  Second  Methods  (Diagonal  and  Perpendic- 
ular Surveying)  will  be  treated  of  under  the  title  "  Chain  Survey- 
ing," in  Part  11.  The  Third  Method  (Polar  Surveying)  will  be 
explained  under  the  titles  "  Compass  Surveymg,"  Part  III,  and 
"  Transit  and  Theodolite  Surveying,"  Part  IV.  The  Fourth  and 
Fifth  Methods  will  be  found  under  their  own  names  of  "  Triangu- 
lar Surveying,"  and  "  Trilinear  Surveying,"  in  Parts  V  and  VI. 

(17)  In  all  the  methods  of  Land  Surveying,  there  are  three 
■tages  of  operation : 

1*^  Measuring  certain  lines  and  angles,  and  recording  them ; 
2<^  Drawing  them  on  paper  to  some  suitable  scale ; 
3°   Calculating  the  content  of  the  surface  surveyed. 
The  three  following  chapters  will  treat  of  each  of  these  topics  ic 
their  turn. 


16  FUNDAMENTAL  OPERATIONS.  [pibx  l 

CHAPTER  II. 

MAKING  THE  MEASUREMENTS. 

(18)  The  Measurements  whicli  are  required  in  Surveying,  ma;y 
be  of  lines  or  of  angles,  or  of  both ;  according  to  the  Method  em- 
ployed     Each  will  be  successively  considered. 

MEASURING  STRAIGHT  LINES. 

(19)  The  lines,  or  distances,  which  are  to  be  measured,  may  be 
either  actual  or  visual. 

Actual  lines  are  such  as  reaUy  exist  on  the  surface  of  the  land 
to  be  surveyed,  either  bounding  it,  or  crossiag  it ;  such  as  fences, 
ditches,  roads,  streams,  &c. 

Visual  lines  are  imaginary  lines  of  sight,  either  temporarily 
measured  on  the  ground,  such  as  those  joining  opposite  corners  o£ 
ft  field ;  or  simply  indicated  by  stakes  at  their  extremities  or  other- 
wise.   K  long,  they  are  "  ranged  out "  by  methods  to  be  given. 

Lines  are  usually  measured  with  chains,  tapes  or  rods,  di- 
vided into  yards,  feet,  links,  or  some  other  unit  of  measurement. 

(20)  Gunter*s  Chain.  This  is  the  measure  most  commonly 
used  in  Land  surveying.  It  is  66  feet,  or  4  rods  long.*  Eighty 
Buch  chains  make  one  mile. 

Fig    8. 


■     ■     •—     «  •  ■     »     ■    ■     * — *\,^ 


zzz> 


It  is  composed  of  one  hundred  pieces  of  iron  wu'e,  or  links,  each 
bent  at  the  end  into  a  ring,  and  connected  with  the  ring  at  the  end 
of  the  next  piece  by  another  ring.  Sometimes  two  or  three  rings 
are  placed  between  the  links.     The  chain  is  then  less  liable  to 

•  This  length  was  chosen  (by  Mr.  Edward  Gunter)  because  10  square  chaim 
of  66  feet  malse  one  acre,  (as  will  be  shown  in  Chapter  IV,)  and  the  computation 
of  areas  is  thus  greatly  facilitated.  For  other  Surveying  pui-poses,  particuiarly 
for  Rail-road  work,  a  chain  of  100  feet  is  preferable.  On  the  U  jited  States 
Coast  Survey,  the  unit  of  measurement  (which  at  some  future  time  will  be  Uie 
universal  one)  is  the  French  Metre,  equsd  to  3.281  feet,  nearly- 


CHAP.    II.] 


Wakinff  the  Measurements. 


17 


twist  and  get  entangled,  or  "  Idnked."  Two  or  more  swivels  are 
also  inserted  in  the  chain,  so  that  it  may  turn  around  without  twist- 
ing. Every  tenth  link  is  marked  by  a  piece  of  brass,  having  one, 
two,  three,  or  four  points,  corresponding  to  the  number  of  tena 
which  it  marks,  counting  from  the  nearest  end  of  the  chain.*  The 
middle  or  fiftieth  link  is  marked  by  a  round  piece  of  brass. 

The  hundredth  part  of  a  chain  is  called  a  link.f  The  great 
advantage  of  this  is,  that  since  links  are  decimal  parts  of  a  chain, 
they  may  be  so  written  down,  5  chains  and  43  hnks  being  5.43 
chains,  and  all  the  calculations  respecting  chams  and  hnks  can  then 
be  performed  by  the  common  rules  of  decimal  Arithmetic.  Each 
Imk  is  7.92  inches  long,  being  =  66  X  12  -^  100. 

The  following  Table  ■\^  ill  be  found  convenient : 


CHAmS   INTO  FEET. 

CILIIXS. 

FEET. 

CHAINS. 

FEET. 

0.01 

0.66 

1.00 

66. 

0.02 

1.32 

2. 

132. 

0.03 

1.98 

3. 

198. 

0.04 

2.64 

4. 

264. 

0.05 

3.80 

5. 

330. 

0.06 

3.96 

6. 

396. 

0.07 

4.62 

7. 

462. 

0.08 

5.28 

8. 

528. 

0.09 

5.94 

9. 

594. 

0.10 

6.60 

10. 

660. 

0.20 

13.20 

20. 

1320. 

0.30 

19.80 

30. 

1980. 

0.40 

26.40 

40. 

2640. 

0.50 

33.00 

50. 

3300. 

0.60 

39.60 

60. 

3960. 

0.70 

46.20 

70. 

4620. 

0.80 

52.80 

80. 

5280. 

0.90 

59.40 

90. 

5940. 

1.00 

66.00 

100. 

6600. 

FEET  INTO  LINKS. 

FEET. 

LINKS. 

FEET. 

LINKS. 

0.10 

0.15 

10. 

15.2 

0.20 

0.30 

15. 

22.7 

0.25 

0.38 

20. 

30.3 

0.30 

0.45 

25. 

37.9 

0.40 

0.60 

30. 

45.4 

0.50 

0.76 

33. 

50.0 

0.60 

0.91 

35. 

53.0 

0.70 

1.06 

40. 

60.6 

0.75 

1.13 

45. 

68.2 

0.80 

1.21 

50. 

75.8 

0.90 

1.36 

55. 

83.3 

1.00 

1.62 

60. 

90.9 

2. 

3.0 

65. 

98.5 

3. 

4.5 

70. 

106.1 

4. 

6.1 

75. 

113.6 

5. 

7.6 

80. 

121.2 

6. 

9.1 

35. 

128.8 

7. 

10.6 

90. 

136.4 

8. 

12.1 

95. 

143.9 

9. 

13.6 

100. 

151.5 

•  To  prevent  the  vei-y  common  mistake,  of  calling  forty,  sixty;  or  thirty,  seventy; 
it  has  been  suggested  to  make  the  11th,  21st,  31st  and  41st  links  of  brais;  which 
would  at  once  show^  on  which  side  of  the  middle  of  the  chain  was  the  doubtful 
mark.    This  would  be  particularly  useful  in  Mining  Surveying. 

t  This  must  not  be  confounded  with  the  pieces  of  wire  which  have  the  same 
name,  since  one  of  them  is  shorter  than  the  "  link"  used  in  calculation,  by  half  a 
ring,  o"  more,  according  to  the  way  in  which  the  chain  is  made. 


18  FUi\DAMEi\TAL  OPERATIONS.  [pakt  i 

To  reduce  links  to  feet,  subtract  from  the  number  of  links  aa 
many  imits  as  it  contains  hundreds ;  multiply  the  remainder  by  2 
and  divide  by  3. 

To  reduce  feet  to  links,  add  to  the  given  number  half  of  itself, 
and  add  one  for  each  hundred  (more  exactly,  for  each  ninety-niae) 
in  the  sum. 

The  chain  is  liable  to  be  lengthened  by  its  rings  being  pulled 
open,  and  to  be  shortened  by  its  links  being  bent.  It  should  there- 
fore be  frequently  tested  by  a  carefully-measured  length  of  QQ  feet, 
set  out  by  a  standard  measure  on  a  flat  surface,  such  as  the  top 
of  a  wall,  or  on  smooth  level  ground  between  two  stakes,  their 
centres  being  marked  by  small  nails.  It  may  be  left  a  little  longer 
than  the  true  length,  since  it  can  seldom  be  stretched  so  as  to  be 
perfectly  horizontal  and  not  hang  in  a  curve,  or  be  drawn  out  in  a 
perfectly  straight  line.*  Distances  measured  with  a  perfectly 
accurate  chain  will  always  and  unavoidably  be  recorded  as  longer 
than  they  reaUy  are.  To  ensure  the  chaui  being  always  strained 
^th  the  same  force,  a  spring,  hke  that  of  a  sprmg-balance, 
is  sometimes  placed  between  one  handle  and  the  rest-  of  the 
chain. 

If  a  line  has  been  measured  with  an  incorrect  chain,  the  true 
length  of  the  line  wiU  be  obtained  by  multiplying  the  number  of 
chains  and  links  in  the  measured  distance  by  100,  and  dividing  by 
the  length  of  the  standard  distance,  as  given  by  measurement  of 
it  with  the  incorrect  chain.  The  proportion  here  employed  is  this : 
As  the  length  of  the  standard  given  by  the  incorrect  chain  Is  to 
the  true  length  of  the  standard.  So  is  the  length  of  the  line  given 
by  the  measurement  To  the  true  length.  Thus,  suppose  that  a 
line  has  been  measured  with  a  certain  chain,  and  found  by  it  to  be 
ten  chains  long,  and  that  the  chain  is  afterwards  foimd  to  have  been 
so  stretched  that  the  standard  distance,  measured  by  it,  appears  to 
be  only  99  links  long.  The  measured  line  is  therefore  longer 
than  it  had  been  thought  to  be,  and  its  true  leng-th  is  obtained 
by  multiplying  ten  by  100,  and  dividing  by  99. 

The  chain  used  by  the  Governtnent  Surveyors  of  France,  which  is  10  Metrea, 
or  about  half  a  Giinter's  chain  in  length,  is  made  from  one-fifth  to  two-fifths  of  an 
inch  longer  than  the  standard.  An  inaccuracy  of  one  five  hundredth  of  its  length 
,=  lij  iuches  on  aGiiiitei-'s  chaixi)  is  the  utmost  allowed  not  to  vitiate  the  survey 


CHAP.  II.]  Making  the  Measurements.  19 

(21)  Pins.  Ten  iron  pins  or  "  arrows,"  usually  accompany  the 
chain.*  They  are  about  a  foot  long,  and  are  made  of  stout  iron 
wire,  sharpened  at  one  end,  and  bent  into  a  ring  at  the  other. 
Pieces  of  red  and  white  cloth  should  be  tied  to  their  heads,  so  that 
they  can  be  easily  found  in  gi'ass,  dead  leaves,  &c. 

They  should  be  strung  on  a  ring,  which  has  a  spring  catch  to 
retain  them.  Their  usual  form  is  shown  in  Fig.  9.  Fig.  9.  Fig.  lo. 
Fig.  10  shows  another  form,  made  very  large,  and   ^  ^\ 

therefore  very  heavy,  near  the  point,  so  that  when  [ 

held  by  the  top  and  dropped,  it  may  fall  vertically. 
The  uses  of  this  will  be  seen  presently. 

(22)  On  u-regular  ground,  two  stout  stakes  about 
six  feet  long  are  needed  to  put  the  forward  chaia- 
man  in  line,  and  to  enable  whichever  of  the  two  is 
lowest,  to  raise  his  end  of  the  chain  in  a  truly  vertical  hne,  and  to 
strain  the  chain  straight. 

A  number  of  long  and  slender  rods  are  also  necessary  for 
"  ranging  out "  lines  between  distant  points,  in  the  manner  to  be 
explained  hereafter ;  in  Part  II,  Chapter  V. 

(23)  How  to  Chain.  Two  men  are  required ;  a  forward  chainr 
main,  and  a  hind  chaui-man ;  or  leader  and  follower.  The  latter 
takes  the  handles  of  the  chain  in  his  left  hand,  and  the  chaui  itself 
m  his  right  hand,  and  throws  it  out  in  the  direction  in  which'  it  is  to 
be  drawn.  The  former  takes  a  handle  of  the  chain  and  one  pin  in 
his  right  hand,  and  the  other  pins  (and  the  staff,  if  used,)  in  his 
left  hand,  and  draws  out  the  chain.  The  follower  then  -walks 
beside  it,  examining  carefully  that  it  is  not  twisted  or  bent.  He 
then  returns  to  its  hinder  end,  which  he  holds  at  the  beginning  of 
the  line  to  be  measured,  puts  his  eye  exactly  over  it,  and,  by  the 
words  "  Right,"  "  Left,"  directs  the  leader  how  to  put  his  staff, 
or  the  pin  which  he  holds  up,  "  in  line,"  so  that  it  may  seem  to 
cover  and  hide  the  flag-staff,  or  other  object  at  the  end  of  the  line. 

*  Eleven  pins  are  sometimes  \3sed,  one  being  of  brass.  Nine  of  iron,  with  four 
or  eight  of  brass,  may  also  be  employed.  Their  uses  are  explained  in  Articles 
(23)  and  (24). 


20  FUNDA.^IEMAL  OPERATIO\S.  [part  i 

The  leader  all  the  while  keeps  the  chain  tightly  stretched,  and  hia 
end  of  it  touching  his  staff.  Every  time  he  moves  the  chain,  he 
should  straighten  it  by  an  undulating  shake.  "\\Tien  the  staff  (or 
pin)  is  at  last  put  "  in  line,"  the  follower  says  "  Down."  The 
leader  then  puts  in  the  single  pin  precisely  at  the  end  of  the  chain, 
and  rephes  "  Down."  The  follower  then  (and  never  before  hearing 
this  signal  that  the  point  is  fixed)  loosens  his  end  of  the  chain, 
retaining  it  in  his  hand.  The  leader  draws  on  the  chain,  making 
a  step  to  one  side  of  the  pin  just  set,  to  avoid  dragging  it  out.  He 
should  keep  his  eye  steadily  on  the  object  ahead,  or,  in  a  hoUow, 
should  line  himself  approximately  by  looking  back.  The  follower 
should  count  his  steps,  so  as  to  know  where  to  look  for  the  pin  in 
high  grass,  &c.  As  he  approaches  the  pin,  he  calls  "  Halt."  On 
reaching  it,  he  holds  the  handle  of  the  chain  against  it,  pressing 
his  knee  against  both  to  keep  the  pin  firm.  He  then,  with  his  eye 
over  the  pin,  "  hues"  the  leader  as  before.  "When  the  "  Down" 
has  been  again  called  by  the  follower,  and  answered  by  the  leader, 
the  former  puUs  out  the  pin  with  the  chain-hand,  and  carries  it  in 
his  other  hand,  and  they  go  on  as  before.*  The  operation  is 
repeated  till  the  leader  has  amved  at  the  end  of  the  line,  or  has 
put  down  all  his  pins. 

When  the  leader  has  put  down  his  tenth  pin,  he  draws  on  the 
chain  its  length  farther,  and  after  being  lined,  puts  his  foot  on  the 
handle  to  keep  it  firm,  and  calls  "  Tally."  The  follower  then 
drops  his  end  of  the  chain,  goes  up  to  the  leader  and  gives  him 
back  all  the  pins,  both  counting  them  to  make  sure  that  none  have 
been  lost.  One  pin  is  then  put  down  at  the  forward  end  of  the 
chain,  and  they  go  on  as  before. 

Some  Surveyors  cause  the  leader  to  caU  "tally"  at  the  tenth 
pin,  and  then  exchange  pins ;  but  then  the  follower  has  only  the 
hole  made  by  the  pin,  or  some  other  indefinite  mark,  to  measure 
from. 

Eleven  pins  are  sometimes  preferred,  the  eleventh  being  of  brass, 
or  otherwise  different  from  the  rest,  and  being  used  to  mark  the 

•  When  a  chain's  length  would  end  in  a  ditch,  pool  of  watei',  &c.  and  the  chain, 
men  are  afraid  of  wetting  their  feet,  they  can  measure  part  of  a  chain,  to  the  edg« 
of  the  water,  then  stretch  the  chain  across  if,  and  thei)  measure  another  poitioi 
of  a  chain,  so  that  with  the  former  poi'rion,  it  may  ;nake  up  a  full  chain. 


CHAP   II. ]  .llakin^  the  Mcasureioents.  21 

end  of  the  eleventh  chain  ;  another  being  substituted  for  it  before 
the  leader  goes  on. 

The  two  chain-men  may  change  duties  at  each  change  of  pins, 
if  thej  are  of  equal  skill,  but  the  more  careful  and  inteUigent  of 
t\vo  laborers  should  generally  be  made  "  follower." 

When  the  leader  reaches  the  end  of  the  line,  he  stops,  and  holds 
his  end  of  the  chain  against  it.  The  follower  drops  his  end  and 
counts  the  links  beyond  the  last  pin,  notuig  carefully  on  which  side 
of  the  "  fifty"  mark  it  comes.  Each  pin  now  held  by  the  follower, 
including  the  one  in  the  ground,  represents  1  chain ;  each  time 
"tally"  has  been  called,  and  the  pins  exchanged,  represents  10 
chains,  and  the  links  just  counted  make  up  the  total  distance. 

(24)  Tallies.  In  chaming  very  long  distances,  there  is  danger 
of  miscounting  the  number  of  "  taUies,"  or  tens.  To  avoid  mis- 
takes, pebbles,  &c.,  may  be  changed  from  one  pocket  into  another 
at  each  change  of  pins ;  or  bits  of  leather  on  a  cord  may  be  slip- 
ped from  one  side  to  the  other ;  or  knots  tied  on  a  string ;  but  the 
best  plan  is  the  following.  Instead  of  ten  iron  pins,  use  nine  iron 
pins,  and  four,  or  eight,  or  ten  pins  of  brass,  or  very  much  longer 
than  the  rest.  At  the  end  of  the  tenth  chain,  the  u'on  pins  being 
exhausted,  a  brass  pin  is  put  down  by  the  leader.  The  follower 
then  comes  up,  and  returns  the  nine  iron  pins,  but  retauis  the  brass 
one,  with  the  additional  advantage  of  having  this  pin  to  measure 
from.  At  the  end  of  the  twentieth  chain,  the  same  operation  is 
repeated ;  and  so  on.  When  the  measurement  of  the  line  is  com- 
pleted, each  brass  pin  held  by  the  follower  counts  ten  chains,  and 
each  iron  pin  one,  as  before. 

(25}  tliaining  on  Slopes.  All  the  distances  employed  in 
Land-surveying  must  be  measui-ed  horizontally,  or  on  a  level ;  for 
reasons  to  be  given  in  chapter  IV.  When  the  ground  slopes,  it  ia 
therefore  necessary  to  make  certain  allowances  or  connections.  If 
the  slope  be  gentle,  hold  the  up-hill  end  of  the  chain  on  the  ground, 
and  raise  the  down-hill  end  tiU  the  chain  is  level.  To  ensure  the 
elevated  end  being  exactly  over  the  desired  spot,  raise  it  along  a 
fcta£f  kept  vertical,  or  drop  a  pm  held  by  the  pcmt  with  the  ring 


22  FUIVDAWEMAL  OPERATIOXS.  [pakt  i 

downwards,  (if  you  have  not  the  heavy  pointed  ones  shown  in  Fig. 
10),  or,  which  is  better,  use  a  plumb-Hne.  A  person  standing 
beside  the  chain,  and  at  a  httle  distance  from  it,  can  best  tell  if  it 
be  nearly  level.  If  the  hill  be  so  steep  that  a  whole  chain  cannot 
be  held  up  level,  use  only  half  or  quarter  of  it  at  a  time.  Great 
care  is  necessary  in  this  operation.  To  measure  down  a  steep  hill, 
Btretch  the  whole  chain  in  line.     Hold  the  Fig.  ii. 

upper  end  fast  on  the  ground.  Raise  up 
the  20  or  30  link-mark,  so  that  that  portion 
of  the  chain  is  level.  Drop  a  plumb-line  or 
pin.  Then  let  the  follower  come  forward 
and  hold  down  that  link  on  this  spot,  and  the  leader  hold  up  an- 
other short  portion,  as  before.  Chaining  do^m  a  slope  is  more 
accurate  than  chaining  up  it,  since  m.  the  latter  case  the  follower 
cannot  easily  place  his  end  of  the  chaia  exactly  over  the  pin. 

(26)  A  more  accurate,  though  more  troublesome,  method,  is  to 
measure  the  angle  of  the  slope ;  and  make  the  proper  allowance 
by  calculation,  or  by  a  table,  previously  prepared.  The  correction 
being  found,  the  chain  may  be  drawn  forward  the  proper  number 
of  Hnks,  and  the  correct  distance  of  the  various  points  to  be  noted 
will  thus  be  obtained  at  once,  without  any  subsequent  calculation 
or  reduction.  If  the  survey  is  made  with  the  Theodolite,  the  slope 
of  the  ground  can  be  measured  directly.  A  "  Tangent  Scale,"  for 
the  same  purpose,  may  be  formed  on  the  sides  of  the  sights  of  a 
Compiss.     It  will  be  described  when  that  instrument  is  explained. 

In  the  following  table,  the  first  column  contains  the  angle  wliich 
the  suiface  of  the  ground  makes  with  the  horizon;  the  second 
column  contains  its  slope,  named  by  the  ratio  of  the  perpendicular 
to  the  base ;  and  the  thu'd,  the  correction  in  links  for  each  chain 
measured  on  the  slope,  i.  e.  the  difference  between  the  hypothenuse, 
which  is  the  distance  measured,  and  the  horizontal  base,  which  ia 
the  distance  desired. 


CHAP.  II.] 


Making  the  measurements. 


2S 


TABLE  FOR  CHAINING  ON  SLOPES. 

ANGLE. 

SLOPE. 

CORRECTION 
IN   LINKS. 

ANGLE. 

SLOPE. 

CORRECTION 
IN   LINKS. 

3° 

Iinl9 

0.14 

130 

lin  41 

2.56 

40 

linl4 

0.24 

140 

lin4 

2.97 

5° 

linll^ 

0.38 

15° 

lin  4 

3.41 

6==> 

lin    9^ 

0.55 

16° 

lin3| 

3.87 

70 

lin    8 

0.75 

170 

lin  31 

4.37 

8- 

1  in    7 

0.97 

18° 

lin  31 

4,89 

90 

lin    61 

1.23 

19° 

lin3 

5.45 

10^ 

lin    6 

1.53 

20° 

lin2| 

6.03 

11° 

1  in    5^ 

1.84 

25° 

lin2 

9.37 

12^ 

1  in    4| 

2.19 

30° 

linl| 

13.40 

(27)  Chaining  is  the  fundamental  operation  in  all  kinds  of  Sur- 
veying. It  has  for  this  reason  been  very  minutely  detailed.  The 
"follower"  is  the  most  responsible  person,  and  the  Sm'veyor  will 
best  ensure  his  accuracy  by  taking  that  place  himself.  If  he  has 
to  employ  inexperienced  laborers,  he  will  do  well  to  cause  them  to 
measui-e  the  distance  between  any  two  points,  and  then  remeasure 
it  in  the  opposite  direction.  The  difference  of  their  two  results 
■will  impress  on  them  the  necessity  of  great  carefulness. 

To  "  do  up"  the  chain,  take  the  middle  of  it  in  the  left  hand, 
and  with  the  right  hand  take  hold  of  the  doubled  chaui  just  beyond 
the  second  hnk ;  double  up  the  two  links  between  your  hands, 
and  continue  to  fold  up  two  double  links  at  a  time,  laying  each  pair 
obhquely  across  the  others,  so  that  when  it  is  all  folded  up,  the 
handles  will  be  on  the  outside,  and  the  chain  will  have  an  hour-glass 
shape,  easy  to  strap  up  and  to  carry. 


(28)  Tape.  Though  the  chain  is  most  usually  employed  for  the 
principal  measurements  of  Surveying,  a  tape-line,  divided  on  one 
side  into  links,  and  on  the  other  into  feet  and  inches,  is  more  con- 
venient for  some  purposes.  It  should  be  tested  very  frequently, 
particularly  after  getting  wet,  and  the  correct  length  marked  on  it 
at  every  ten  feet.     A  "  Metallic  Tape,"  less  hable  to  stretch,  lias 


24  FUNDAMENTAL  OPERATlOINS.  [pakt  i. 

been  recently  manufactured,  in  fvliicli  fine  vrires  form  its  warp. 
When  the  tape  is  being  wound  up,  it  should  be  passed  between  two 
fingers  to  prevent  its  t-wisting  in  the  box,  which  would  make  it 
necessary  to  unscrew  its  nut  to  take  it  out  and  untwist  it.  While 
in  use,  it  should  be  made  portable  by  being  folded  up  by  arm's 
lengths,  instead  of  being  wound  up. 

(29)  Substitutes  for  a  chain  or  a  tape,  may  be  found  in  leather 
driving  lines,  marked  off  with  a  carpenter's  rule,  or  in  a  cord  knot- 
ted at  the  length  of  every  link.  A  well  made  rope,  (such  as  a 
"  patent  wove  line,"  woven  circularly  with  the  strands  always 
straight  in  the  line  of  the  strain),  when  once  well  stretched,  wetted 
and  allowed  to  dry  with  a  moderate  strain,  wUl  not  vary  from  a 
chain  more  than  one  foot  in  two  thousand,  if  carefully  used. 

(30)  Rods*  When  unusually  accurate  measurements  are  re- 
quired, rods  are  employed.  They  may  be  of  well  seasoned  wood, 
of  glass,  of  iron,  &c.  They  must  be  placed  in  line  very  carefully 
end  to  end ;  or  made  to  coincide  in  other  ways  ;  as  wUl  be  explain- 
ed in  Part  V,  under  the  title  of  "  Triangular  Surveying,"  in 
which  the  peculiai'ly  accurate  measurement  of  one  line  is  required, 
as  all  the  others  are  founded  upon  it. 

(31)  Pacing,  Sound,  and  other  approximate  means,  may  be 
used  for  measurmg  the  length  of  a  Hue.  They  will  be  discussed, 
in  Part  IX,    The  Stadia  is  described  in  Art.  (375.) 

(32)  A  Perambulator,  or  "  Measuring  Wheel,"  is  sometimes 
used  for  measuring  distances,  particularly  Roads.  It  consists  of  a 
wheel  which  is  made  to  roll  over  the  ground  to  be  measured,  and 
whose  motion  is  communicated  to  a  series  of  toothed  wheels  within 
the  machine.  These  wheels  are  so  proportioned,  that  the  index 
wheel  registers  their  revolutions,  and  records  the  whole  distance 
passed  over.  If  the  diameter  of  the  wheel  be  31^  inches,  the  cu*- 
cumference,  and  therefore  each  revolution,  will  be  8|  feet,  or  half 
a  rod.  The  roughnesses  of  the  road  and  the  slopes  necessarily 
cause  the  registered  distances  to  exceed  the  true  measure. 


CHAP.  111.]  Making  the  Measurements.  25 

MEASURING  ANGLES. 

(33)  The  angle  made  bj  any  two  lines,  that  is,  the  difference 
of  their  directions,  is  measured  by  vai'ious  instruments,  consisting 
essentially  of  a  circle  divided  into  equal  parts,  with  plain  sights,  or 
telescopes,  to  indicate  the  directions  of  the  two  Imes. 

As  the  measurement  of  angles  is  not  requu-ed  for  "  Chain  Sur- 
veying," which  is  the  first  Method  to  be  discussed,  the  considera- 
tion of  this  kind  of  measurement  will  be  postponed  to  Part  III. 

NOTING   THE    MEASUREMENTS. 

(34)  The  measurements  which  have  been  made,  whether  of 
lines,  or  of  angles,  require  to  be  very  carefully  noted  and  recorded. 
Clearness  and  bre^nty  are  the  points  desu*ed.  Different  methods 
of  notation  are  requii-ed  for  each  of  the  systems  of  surveying  which 
are  to  be  explained,  and  will  therefore  be  ^ven  in  their  appropriate 
places. 


CHAPTER  III. 

DRAWING  THE  MIP. 

(35)  A  Map  of  a  survey  represents  the  lines  wliich  bound  the 
surface  surveyed,  and  the  objects  upon  it,  such  as  fences,  roads, 
rivers,  houses,  woods,  hills,  &c.,  in  their  true  relative  dimensions 
and  positions.  It  is  a  miniatui-e  copy  of  the  field,  farm,  &c.,  as  it 
would  be  seen  by  an  eye  moving  over  it ;  or  as  it  would  appear,  if 
from  every  point  of  its  irregular  surface,  plumb  Knes  were  dropped 
to  a  level  surface  under  it,  forming  what  is  called  in  geometrical 
language,  its  liorizontal  projection. 

(36)  Platting'.  A  plat  of  a  survey  is  a  skeleton,  or  outline 
map.  It  is  a  figure  "  similar"  to  the  onguial,  ha-sdng  all  its  angles 
equal,  and  its  sides  projwrtional.  Every  inch  on  it  represents  a 
foot,  a  yard,  a  rod,  a  mile,  or  some  other  length,  on  the  ground ; 


26  FUNDAMENTAL  OPERATIONS.  [part  i. 

all  the  measured  distances  being  diminished  in  exactly  the  samo 
ratio. 

Platting  is  repeating  on  faper,.  to  a  smaller  scale,  the  mea 
surements  which  have  been  made  on  the  ground. 

Its  various  operations  may  therefore  be  reduced,  in  accordance 
with  the  principles  established  in  the  Fig.  12 

jEirst  chapter,  to  two,  viz :  drawing 
a  straight  line  in  a  given  direction 
and  of  a  given  length ;  and  describ-  ^ 
ing  an  arc  of  a  circle  with  a  radius 
whose  length  is  also  given.  The 
only  instruments  absolutely  necessary  for  this,  are  a  straight  ruler, 
and  a  pair  of  "  dividers,"  or  "  compasses."  Others,  however,  are 
often  convenient,  and  wUl  be  now  briefly  noticed. 

(37)  Straight  Lines.    These  are  usually  drawn  by  the  aid  of  a 

straightrodged  iniler.  But  to  obtain  a  very  long  straight  line  upon 
paper,  stretch  a  fine  silk  thread  between  any  two  distant  points, 
and  mark  in  its  hne  various  points,  near  enough  together  to  be 
afterwards  connected  by  a  common  ruler.  The  thread  may  also 
be  blackened  with  burnt  cork,  and  snapped  on  the  paper,  as  a 
carpenter  snaps  his  chalk  line. ;  but  this  is  liable  to  inaccuracies, 
from  not  raising  the  line  vertically. 

(38)  Arcs.  The  arcs  of  circles  used  in  fixing  the  position  of  a 
point  on  paper,  are  usually  described  with  compasses,  one  leg  of 
which  carries  a  pencil  point.  A  convenient  substitute  is  a  strip 
of  pasteboard,  through  one  end  of  which  a  fine  needle  is  thrust  into 
the  given  centre,  and  through  a  hole  in  which,  at  the  desired  di* 
tance,  a  pencil  pomt  is  passed,  and  can  thus  describe  a  circle  about 
the  centre,  the  pasteboard  keepmg  it  always  at  the  proper  distance. 
A.  string  is  a  still  readier,  but  less  accurate,  instrument. 

(39)  Parallels.  The  readiest  mode  of  drawing  parallel  lines 
is  by  the  aid  of  a  triangular  piece  of  wood  and  a  ruler.     Let  AB 


CHAP.    II.] 


Dra^Finff  the  Wap. 


27 


be  the  line  to  which  a  parallel  is  to 

be  drawn,  and  C  the  point  through 

which  it  must  pass.     Place   one 

side  of  the  triangle   against  the 

line,  and  place  the  ruler  against 

another  side  of  the  triangle.     Hold 

the  ruler  firm  and  immovable,  and 

slide  the  triangle  along  it  till  the  side  of  the  triangle  which  had  coio- 

cided  with  the  given  line,  passes  through  the  given  point.     This 

side  will  then  be  parallel  to  that  given  line,  and  a  line  drawn  by 

it  will  be  the  line  required. 

Another  easy  method  of  drawing  parallels,  is  by  means  of  a  T 
square,  an  instrument  very  valuable  for  many  other  purposes.  It 
is  nothing  but  a  ruler  let  into  a  thicker  piece  of  wood,  very  truly 
at  right  angles  to  it.  For  this  use  of  it,  one  side  of  the  cross-piece 
must  be  even,  or  "  flush,"  with  the  ruler.  To  use  it,  lay  it  on  the 
paper  so  that  one  edge  of  the  Fig.  14. 

ruler  coincides  with  the  given  line 
AB.  Place  another  ruler  against 
the  cross-piece,  hold  it  firm,  and 
slide  the  T  square  along,  till  its 
edge  passes  through  the  ^ven 
point  C,  as  shown  by  the  lower 
part  of  the  figure.  Then  draw 
by  this  edge  the  desired  line  paral- 
lel to  the  given  hne. 

(40)  Perpendiculars,  These  may  be  drawn  by  the  varioua 
problems  given  in  Geometry,  but  more  readily  by  a  triangle  which 
has  one  i-ight  angle.  Place  the  longest 
side  of  the  triangle  on  the  given  line, 
and  place  a  ruler  against  a  second  side 
of  the  ti-iangle.  Hold  the  ruler  fast, 
and  turn  the  triangle  so  as  to  bring  its 
tlurd  side  against  the  ruler.  Then  will 
the  long  side  be  perpendicular  to  the 


28  FUNDAMEMAL  OPERATIONS.  [part  i 

given  line.  By  sliding  the  triangle  along  the  ruler,  it  may  ba 
used  to  draw  a  perpendicular  from  any  point  of  the  line,  or  from 
any  point  to  the  line. 

(41)  Ansfles.  These  are  most  easily  set  out  with  an  instru- 
ment called  a  Protractor,  usually  a  semi-cu-cle  of  brass.  But  the 
description  of  its  use,  and  of  the  other  and  more  accurate  modes 
of  laying  off  angles,  will  be  postponed  till  they  are  needed  in  Part 
HI,  Chapter  IV. 

(42)  Drawlns?  to  Scale.  The  operation  of  drawing  on  paper 
lines  whose  length  shall  be  a  half,  a  quarter,  a  tenth,  or  any  other 
fraction,  of  the  Unes  measured  on  the  ground,  is  called  "  Drawing 
to  Scale." 

To  set  off  on  a  line  any  given  distance  to  any  required  scale, 
determine  the  number  of  chains  or  links  wliich  each  division  of 
the  scale  of  equal  parts  shall  represent.  Divide  the  given  distance 
by  this  number.  The  quotient  wUl  be  the  number  of  equal  parts 
to  be  taken  in  the  dividers  and  to  be  set  off. 

For  example,  suppose  the  scale  of  equal  parts  to  be  a  common 
carpenter's  rule,  divided  into  inches  and  eighths.  Let  the  given 
distance  be  twelve  chains,  which  is  to  be  drawn  to  a  scale  of 
two  chauis  to  an  inch.  Then  six  inches  will  be  the  distance  to  be 
set  off.  If  the  given  distance  had  been  twelve  chains  and  seventy 
five  Unks,  the  distance  to  be  set  off  would  have  been  six  inches 
and  three-eighths,  since  each  eighth  of  an  inch  represents  25  links. 

If  the  desired  scale  were  three  chains  to  an  inch,  each 
eighth  of  an  inch  would  represent  37|  links ;  and  the  distance 
of  1275  links  would  be  represented  by  thirty-four  eighths  of  an  inch, 
or  4^  inches. 

A  similar  process  wiU  give  the  correct  length  to  be  set  off  for 
any  distance  to  any  scale. 

If  the  scale  used  had  been  divided  into  inches  and  tenths,  as  is 
much  the  most  convenient,  the  above  distances  would  have  become 
on  the  former  scale  6^^^^  inches,  or  nearly  6j\  inches  ;  and  on  the 
latter  scale  4y^o^^  inches,  coming  midway  between  the  2d  and  8d 
tenth  of  an  inch. 


CHAP.  III.]  Drawing  the  Map.  29 

(43)  Conversely,  to  find  the  real  length  of  a  line  drawn  on 
paper  to  any  known  scale,  reverse  the  preceding  operation.  Take 
the  length  of  the  line  in  the  dividers,  apply  it  to  the  scale,  and 
count  how  many  equal  parts  it  includes.  Multiply  their  number 
by  the  number  of  chains  or  links  which  each  represents,  and  the 
product  will  be  the  desired  length  of  the  Hne  on  the  ground. 

This  operation  and  the  preceding  one  are  greatly  facihtated  by 
the  use  of  the  scales  to  be  described  in  Art.  C48) 

(44)  Scales.  The  choice  of  the  scale  to  which  a  plat  should  be 
drawn,  that  is,  how  many  times  smaller  its  lines  shall  be  than  those 
which  have  been  measured  on  the  ground,  is  determined  by  several 
considerations.  The  chief  one  is,  that  it  shall  be  just  large  enough 
to  express  clearly  all  the  details  which  it  is  desirable  to  know.  A 
Farm  Survey  would  require  its  plat  to  show  every  field  and  build- 
ing. A  State  Survey  would  show  only  the  towns,  rivers,  and  lead- 
ing roads.  The  size  of  the  paper  at  hand  wiU  also  limit  the  scale 
to  be  adopted.  If  the  content  is  to  be  calculated  from  the  plat, 
that  will  forbid  it  to  be  less  than  3  chains  to  1  inch. 

Scales  are  named  in  various  ways.  They  should  always  be 
expressed  fractionally ;  i.  e.  they  should  be  so  named  as  to  indicate 
what  fractional  part  of  the  real  line  measured  on  the  ground,  the 
representative  line  drawn  on  the  paper,  actually  is.  ^\Tien  custom 
requires  a  different  way  of  naming  the  scale,  both  should  be  given. 
It  would  be  stiU  better,  if  the  denominator  could  always  be  some 
power  of  10,  or  at  least  some  multiple  of  2  and  5,  such  as  5^01 
TWO?  2A0)  "2^07)  ^^-  For  convenience  in  printing,  these  may  be 
written  thus :  1 :  500,  1 :  1000,  1 :  2000,  1 :  2500,  &c. 

Plats  of  Farm  Surveys  are  usually  named  as  being  so  many 
chains  to  an  inch. 

Maps  of  Surveys  of  States  are  generally  named  as  being  made 
to  a  scale  of  so  many  miles  to  an  inch. 

Maps  of  Rail-road  Surveys  are  said  to  be  so  many  feet  to  an 
inch,  or  so  many  inches  to  a  mile. 


(45)  Farm  Surveys.     If  these  are  of  small  extent,  two  chains 

1 

1564 


to  one  inch  (which  is  =  =  -—  =  1 :  1584)  is  convenient 


so 


FUNDAMEIVTAL  OPERATIOAS. 


[part  l 


A  scale  of  one  chain  to  one  inch  (1 :  792)  is  useful  for  plans  of  build 
ings.  Three  chains  to  one  inch  (1 :  2376)  is  suitable  for  larger 
farms.  It  is  the  scale  prescribed  by  the  EngUsh  Tithe  Commis- 
sioners for  their  first  class  maps. 

In  France,  the  Cadastre  Surveys  are  lithographed  on  a  scale 
about  equivalent  to  this,  bemg  1 :  2500.  The  original  plans  are 
drawn  to  a  scale  of  1:5000.  Plans  for  the  division  of  propcHy 
are  made  on  the  former  scale.  When  the  district  exceeds  3000 
acres,  the  scale  is  1 :  10,000.  When  it  exceeds  7,500  acres,  the 
scale  is  1:20,000.     A  common  scale  in  France  for  small  surveys 

is  1 :1000;  about  1^  chains  to  1  inch. 

•Fig.  16. 
OISTE  ACRE 


ON    SCALE    OF    1    CHAIN    TO    1  INCH. 


■^yir^nJiS^S    TO  1  TSCK. 


■^y 


-^y- 


-&*- 


-4 W — »- 


6    »  11) 
1  io»*  1 

The  choice  of  the  most  suitable  scale  for  the  plat  of  a  farm  sui- 
vey,  may  be  facilitated  by  the  Figure  given  above,  which  shows 
the  actual  space  occupied  by  one  acre,  (the  customary  unit  of  land 
measure),  laid  out  in  the  form  of  a  square,  on  maps  drawn  to  the 
various  scales  named  in  the  fisrure. 


CHAP.  III.]  Drawing^  the  .Hap.  31 

(46)  State  Surveys.  On  these  surveys,  smaller  scales  are 
necessarily  employed. 

On  the  admirable  United  States  Coast  Survey,  all  the  scales 
are  expressed  fractionally  and  decimally.  "  The  surveys  are 
generally  platted  originally  on  a  scale  of  one  to  ten  or  twenty  thou- 
sand, but  in  some  instances  the  scale  is  larger  or  smaller. 

These  original  surveys  are  reduced  for  engraving  and  pubUca- 
tion,  and  when  issued,  are  embraced  in  three  general  classes.  1°, 
small  Harbor  charts ;  2°,  charts  of  Bays,  Sounds,  and  S*^,  of  the 
Coast  General  Charts. 

The  scales  of  the  first  class  vary  from  1:10,000  to  1:60,000, 
according  to  the  nature  of  the  Harbor  and  the  different  objects  to 
be  represented. 

Where  there  are  many  shoals,  rocks,  or  other  objects,  as  in 
Nantucket  Harbor  and  Hell-Gate,  or  where  the  importance  of  the 
harbor  makes  it  necessary,  a  larger  scale  of  1:5,000,  1:10,000, 
and  1:20,000  is  used.  But  where,  from  the  size  of  the  harbor, 
or  its  ease  of  access,  a  smaller  one  will  point  out  every  danger  with 
sufficient  exactness,  the  scales  of  1:40,000  and  1:60,000  are 
used,  as  in  the  case  of  New-Bedford  Harbor,  Cat  and  Ship  Island 
Harbor,  New-Haven,  &c. 

The  scale  of  the  second  class,  in  consequence  of  the  large  areas 
to  be  represented,  is  usually  fixed  at  1 :  80,000,  as  in  the  case  of 
New- York  Bay,  Delaware  Bay  and  River.  Preliminary  charts, 
however,  are  issued,  of  various  scales  from  1 :  80,000  to  1 :  200,000. 

Of  the  third  class,  the  scale  is  fixed  at  1:400,000,  for  the 
General  Chart  of  the  Coast  from  Gay  Head  to  Cape  Henlopen, 
although  considerations  of  the  proximity  and  importance  of  points 
on  the  coast,  may  change  the  scales  of  charts  of  other  portions  of 
our  extended  coast."* 

The  National  Survey  of  Grreat  Britain  is  called,  from  the  corps 
employed  on  it,  the  "  Ordnance  Survey." 

The  "  Ordnance  Survey"  of  the  southern  counties  of  England 
was  platted  on  a  scale  of  2  inches  to  1  mile,  (1:31,680),  and 
reduced  for  publication  to  that  of  one  inch  to  a  mile,  (1:63,860). 
The  scale  of  6  inches  to  a  mile  (1 :  10,560)  was  adopted  for  the 

*  Communicated  from  the  U.  S.  Coast  Survey  office 


32  FIJNDAMEKTAL  OPERATIOXS.  [part  i 

northern  counties  of  England  and  for  the  southern  counties  of  Scot- 
land. The  same  scale  was  employed  for  platting  and  engraving  in 
outhne  the  "  Ordnance  Survey"  of  Ireland.  But  a  map  on  a 
scale  of  1  inch  to  1  mile  (1: 63,360)  is  about  to  be  pubhshed,  the 
former  scale  rendering  the  maps  too  unwieldy  and  cumbrous  for 
consultation. 

The  Ordnance  Survey  of  Scotland  was  at  first  platted  on  a  scale 
of  sis  inches  to  one  mile,  (1:10,660).  That  scale  has  since  been 
abandoned,  and  it  is  now  platted  on  a  scale  of  two  inches  to  1  mile, 
(1 :  31,680),  and  the  general  maps  are  made  to  only  half  that  scale. 

The  Ordnance  Survey  scale  for  the  maps  of  London  and  other 
large  towns,  is  5  feet  to  1  mile,  (1 :  1056),  or  1|  chains  to  one  inch. 

In  the  "Surveys  under  the  Public  Health  act''  of  England, 
the  scale  for  the  general  plan  is  two  feet  to  one  mile,  (1:2,640)  ; 
and  for  the  detailed  plan,  ten  feet  per  mile,  (1:528),  or  two-thirds 
of  a  chain  per  inch. 

The  Government  Survey  of  France  is  platted  to  a  scale  of 
1:20,000.  Copies  are  made  to  1:40,000;  and  the  maps  are 
engi'aved  to  a  scale  of  1: 80,000,  or  about  |  inch  to  1  mile. 

Cassini's  famous  map  of  France  was  on  a  scale  of  1 :  86,400. 

The  French  War  Department  employ  the  scales  of  1:10,000  ; 
1:20,000;  1:40,000;  and  1:80,000;  for  the  topography  of 
France. 

(47)  Rail-road  Surveys,  For  these  the  New-York  General 
Rail-road  Law  of  1850  directs  the  scale  of  maps  which  are  to  be 
filed  in  the  State  Engineer's  Office,  to  be  five  h\mdred  feet  to  one- 
tenth  of  a  foot,  (=  1 :  5000.) 

For  the  New- York  Canal  Maps  a  scale  of  2  chains  to  1  mch 
(1 :  1584)  is  employed. 

The  Parliamentary  "  standmg  orders"  prescribe  the  plans  of 
Rail-roads,  prepared  for  Parliamentary  purposes,  to  be  made  on  a 
scale  of  not  less  than  4  inches  to  the  mile,  (1 :  15840)  :  and  the 
enlarged  portions  (as  of  gardens,  court-yards,  &c.)  to  be  on  a  scale 
not  smaller  than  400  feet  to  the  inch,  (1 :  4800.)  Accordingly 
the  practice  of  English  Railway  Engineers  is  to  draw  the  whole 
plan  to  a  scale  of  6  chains,  or  396  feet  to  the  inch,  (1 :  4752)  as 
being  just  within  the  Parliamentary  limits. 


CHAP,   m.] 


Drawing  the  Map. 


In  France,  the  Engineers  of  "  Bridges  and  Roads"  (Corps  des 
Fonts  et  Chaussees)  employ  for  the  general  plan  of  a  road  a  scale 
of  1 :  5000,  and  for  appropriations  1 :  500. 

(48)  In  the  United  States  Engineer  service,  the  following  scalee 
are  prescribed : 

General  plans  of  buildings,  1  inch  to  10  feet,  (1;120). 

Maps  of  ground,  with  horizontal  curves  one  foot  apart,  1  inch  to  50  feet,  (1 :600  V 

Topographical  maps,  one  mile  and  a  half  square,  2  feet  to  one  mile,  (1 :  2,640). 

Do.  comprising  three  miles  square,  1  foot  to  one  mile,  (1 : 5,280). 

Do.  between  four  and  eight  miles  square,  6  inches  to  one  mile,  (1 :  10,560). 

Do.  comprising  nine  miles  square,  4  inches  to  one  mile,  (1 :  15,840). 

Maps  not  exceeding  24  miles  square,  2  inches  to  one  mile,  (1 :  31,680). 

Maps  compi-ising  50  miles  square,  1  inch  to  one  mile,  (1 :  63,360). 

Maps  comprising  100  miles  square,  ^  inch  to  one  mile,  (1 :  126,720.) 

Surveys  of  Roads,  Canals,  &c.,  1  inch  to  50  feet,  (1 :  600). 

(49)  The  most  convenient  scales  of  equal  parts  are  those  of  box- 
wood, or  ivory,  which  have  a  fiducial  or  feather  edge,  along  which 
they  are  divided,  so  that  distances  can  be  at  once  marked  off  from 
this  edge,  without  requiring  to  be  taken  off  with  the  dividers  ;  or 
the  length  of  a  given  line  can  be  at  once  read  off.  Box-wood  is 
preferable  to  ivory  as  much  less  liable  to  warp,  or  to  vary  in  length 
with  changes  in  the  moisture  in  the  air. 

The  student  can,  however,  make  for  himself  platting  scales  oi 
drawing  paper,  or  Bristol  board.  Cut  a  straight  strip  of  this  mate- 
rial, about  an  inch  wide.     Draw  a  line  through  its  middle,  and  set 


Fig.   17. 

'^ 

o 

It 

1 

\ 

< 

> 

\ 

> 

r 

off  on  it  a  number  of  equal  parts,  each  representing  a  chain  to  the 
desired  scale.  Sub-divide  the  left  hand  division  into  ten  equal 
parts,  each  of  which  will  therefore  represent  ten  links  to  this  scale. 
Through  each  pomt  of  division  on  the  central  line,  draw  (with 
the  T  square)  perpendiculars  extending  to  the  edges,  and  the 
Bcale  is  made.  It  explains  itself.  The 'above  figure  is  a  scale  of 
2  chains  to  1  mch.     On  it  the  distance  220  links  would  extend 


84 


FUNDAMEIVTAL  OPERATIONS. 


[part  1, 


between  the  arrow-heads  above  the  line  in  the  figure  ;  560  links 
extends  between  the  lower  arrow-heads,  &c. 

A  paper  scale  has  the  great  advantage  of  varying  less  from 
a  plat  which  has  been  made  by  it,  in  consequence  of  changes 
in  the  weather,  than  any  other.  The  mean  of  many  trials 
showed  the  difference  between  such  a  scale  and  drawing  paper, 
when  exposed  alternately  to  the  damp  open  atmosphere,  and  to  the 
air  of  a  warm  dry  room,  to  be  equal  to  .005,  while  that  between 
box-wood  scales  and  the  paper  was  .012,  or  nearly  2i  times  as 
much.     The  difference  with  ivory  would  have  been  even  greater. 

Some  of  the  more  usual  platting  scales  are  here  given  in 
their  actual  dimensions. 

In  these  five  figures,  different  methods  of  drawing  the  scales 
have  been  given,  but  each  method  may  be  appHed  to  any  scale. 
The  first  and  second,  being  the  most  simple,  are  generally  the  best. 
In  the  third  the  subdivisions  are  made  by  a  diagonal  line  :  the 
distances  between  the  various  pairs  of  arrow  heads,  beginning  with 
the  uppermost,  are,  respectively,  310,  540,  and  270  links. 

Fig.   18.     Scale  of  1  chain  to  1  inch. 
I  O  1  2 

lioo 


Mil 


Fig.  19.     Scale  of  2  chains  t-o  1  inch. 


^ 


"ST 


B 


Fig.  20.     Scale  of  3  chains  to  1  inch. 
12.3456  78 


lO/. 

\ 

\ 

«v« 

I 

ritf 

1 

J«' 

1 

1 

\ 

1 

( 

In  the  fourth  figure  the  distances  between  the  arrow  heads  are 
respectively  310,  270,  and  540  links. 

Fig.  21.     Scale  of  4  chains  to  1  inch. 
O        1         2        3        4         5        6         7        8        9       10       n       12       13 


\    1 

^ 

\ 

1 

TT 

- 

? 

' 

" 

] 

\ 

: 

1 

■< 

poV-^iio 

/ 

\ 

iM 

J 

CHAP.    III.] 


Drawing  the  Map. 


35 


In  the  fifth  figure  the  scale  of  5  chains  to  1  mch  is  subdivided 
diagonaUj  to  onlj  every  quarter  chain,  or  25  links.  The  distance 
between  the  upper  pair  of  arrow-heads  on  it  is  12^  chains,  or  12.25 ; 
between  the  next  pair  of  arrow-heads,  it  is  6.50  ;  and  between  the 
lower  pair,  14.75. 


Fig.  22.     Scale  of  5  chains  to  1  inch. 


10 


^^ 


±r 


-=r^- 


A  diagonal  scale  for  dividmg  an  inch,  or  a  half  inch,  mto  100 
equal  parts,  is  found  on  the  "  Plain  scale"  in  every  case  of  instru- 
ments. 


(5i)  Vernier  Scale.  This  is  an  ingenious  substitute  for  the 
diagonal  scale.  The  one  gi^-en  m  the  following  figure  divides  an 
inch  into  100  equal  parts,  and  if  each  inch  be  supposed  to  represent 


a  chain,  it  gives  single  finks. 


Fig.  23. 


iOO 


1(1)0 


2(|)0 


Make  a  scale  of  an  inch  divided  into  tenths,  as  in  the  upper 
scale  of  the  above  figure.  Take  in  the  dividers  eleven  of  these 
divisions,  and  set  ofi'  this  distance  from  the  0  of  the  scale  to  the 
left  of  it.  Divide  the  distance  thus  set  off  into  10  equal  parts. 
Each  of  them  will  be  one  tenth  of  eleven  tenths  of  one  iinch ;  i.  e. 
eleven  hundredths,  or  a  tenth  and  a  hundredth,  and  the  first  di- 
vision on  the  short,  or  vernier  scale,  will  overlap,  or  be  longer  than 
the  first  division  on  the  long  scale,  by  just  one  hundredth  of  an 
inch ;  the  second  division  will  overlap  ttvo  hundredths,  and  so  on. 
The  principle  will  be  more  fully  developed  in  treating  of"  Verniers," 
Part  IV,  Chapter  II. 

Now  suppose  we  wish  to  take  off  from  this  scale  275  hundredths 
of  an  inch.  To  get  the  last  figure,  we  must  take  five  divisions  on 
the  lower  scale,  which  will  be  55  hundredths,  for  the  reason  just 
given.     220  will  remam  which  are  to  be  taken  from  the  upper 


S6 


FUIVDAMENTAL  OPERATIONS. 


[part  J 


Bcale,  and  the  entire  number  will  be  obtained  at  once  bj  extending 
the  dividers  between  the  arrow-heads  in  the  figure  from  220  on  the 
upper  scale  (measuring  along  its  lower  side)  to  55  on  the  lower  scale, 
254  would  extend  from  210  on  the  upper  scale  to  44  on  the  lower, 
318  would  extend  from  230  on  the  upper  scale  to  88  on  the  lower. 
Always  begin  then  with  subtracting  11  times  the  last  figure  from 
the  given  number ;  find  the  remainders  on  the  upper  scale,  and 
the  number  subtracted  on  the  lower  scale. 

(51)  A  plat  is  sometimes  made  by  a  ncminally  reduced  scale 
in  the  following  manner.  Suppose  that  the  scale  of  the  plat  is  to 
be  ten  chains  to  one  inch,  and  that  a  diagonal  scale  of  inches,  divided 
into  tenths  and  'hundredths,  is  the  only  one  at  hand.  By  dividing 
all  the  distances  by  ten,  this  scale  can  then  be  used  without  any 
further  reduction.  But  if  the  content  is  measured  from  the  plat 
to  the  same  scale,  in  the  manner  explained  in  the  next  chapter,  the 
result  must  be  multiplied  by  10  times  10.  Tliis  is  called  by  old 
Surveyors  "  Raising  the  scale,"  or  "  Restoring  true  measure." 


(52)  Sectoral  Scales.  The  Sector,  (called  by  the  French 
"  Compass  of  Proportion"),  is  an  instrument  sometimes  convenient 
for  obtaining  a  scale  of  equal  parts.  It  is  in  two  portions,  turning 
on  a  hinge,  like  a  carpenter's  pocket  rule.  It  contains  a  great 
number  of  scales,  but  the  one  intended  for  this  use  is  lettered  at  its 
ends  L  in  Enghsh  instruments,  and  consists  of  two  lines  running 
from  the  centre  to  the  ends  of  the  scale,  and  each  di^dded  into  ten 
equal  parts,  each  of  which  is  again  subdivided  into  10,  so  that  eacli 
leg  of  the  scale  contains  100  F'g-  24. 

equal  parts.  To  illustrate 
its  use,  suppose  that  a  scale 
of  7  chains  to  1  inch  is  re- 
quu'ed.  Take  1  inch  in  the 
dividers,  and  open  the  sec- 
tor till  this  distance  will  just 
reach  from  the  7  on  one  leg 
to  the  7  on  the  other.  The 
sector  is  then  "set"  for  this 


CHAP.  Ill]  Drawing  the  Map.  87 

scale,  and  the  angle  of  its  opening  must  not  be  again  changed. 
Now  let  a  distance  of  580  links  be  required.  Open  the  div-idera 
till  the  J  reach  from  58  to  58  on  the  two  legs,  as  in  the  dotted  line 
in  the  figure,  and  it  is  the  required  distance.  Again,  suppose  that 
a  scale  of  2|  chains  to  one  inch  is  desired.  Open  the  sector  so 
that  1  inch  shall  extend  from  25  to  25.  Any  other  scale  may  be 
obtained  in  the  same  manner. 

Conversely,  the  length  of  any  known  line  to  any  desired  scale 
can  thus  be  readily  determined. 

(53)  Whatever  scale  may  be  adopted  for  platting  the  survey,  it 
should  be  drawn  on  the  map,  both  for  convenience  of  reference, 
and  in  order  that  the  contraction  and  expansion,  caused  by  changes 
in  the  quantity  of  moisture  in  the  atmosphere,  may  affect  the  scale 
and  the  map  ahke.  When  the  di-awing  paper  has  been  wet  and 
glued  to  a  board,  and  cut  off  when  the  map  is  completed,  its  con- 
tractions have  been  found  by  many  observations  to  average  from 
one-fourth  to  one-half  per  cent,  on  a  scale  of  3  chains  to  an  inch, 
(1:2376),  wliich  would  therefore  require  an  allowance  of  from 
one-half  perch  to  one  perch  per  acre. 

A  scale  made  as  directed  in  Art.  (49),  if  used  to  make  a  plat 
on  imstretched  paper,  and  then  kept  with  the  plat,  will  answer 
nearly  the  same  purpose. 

Such  a  scale  may  be  attached  to  a  map,  by  slipping  it  through 
two  or  three  cuts  in  the  lower  part  of  the  sheet,  and  Avill  be  a  very 
convenient  substitute  for  a  pair  of  dividers  in  measm-ing  any  dis- 
tance upon  it. 

(.54)  Scale  omitted.  It  may  be  required  to  find  the  miknown 
acale  to  which  a  given  map  has  been  drawn,  its  supei-ficial  content 
being  known.  Assume  any  convenient  scale,  measure  the  lines 
of  the  map  by  it,  and  find  the  content  by  the  methods  to  be  given 
in  the  next  chapter,  proceeding  as  if  the  assumed  scale  were  the 
true  one.  Then  make  this  proportion,  founded  on  the  geometrical 
principle  that  the  areas  of  similar  figures  are  as  the  squares  of  their 
corresponding  sides :  As  the  content  found  Is  to  the  given  content 
So  is  the  square  of  the  assumed  scale  To  the  square  of  the  true  scalei 


M 


FUNDAMENTAL  OPERATIONS. 


PART  I 


CHAPTER  IV. 

CALCULATING  THE  CONTENT. 
(55)  The  Content  of  a  piece  of  ground  is  !ts  superficial  area^ 
or  the  number  of  square  feet,  yards,  acres,  or  miles  which   it 
contains. 


(56)  Horizontal  Measurement.  AU  ground,  however  inclined 
or  uneven  its  sui'face  may  be,  should  be  measured  horizontally,  or 
as  if  brought  do^vn  to  a  horizontal  plane,  so  that  the  surface  of  a 
hill,  thus  measured,  would  give  the  same  content  as  the  level  base 
on  v.'hich  it  may  be  supposed  to  stand,  or  as  the  figure  which  would 
be  formed  on  a  level  surface  beneath  it  by  dropping  plumb  lines 
from  every  point  of  it. 

This  method  of  procedure  is  required  for  both  Geometrical  and 
Social  reasons. 

Cf-eometricaUi/,  it  is  plain  that  this  horizontal  measurement  is 
absolutely  necessary  for  the  purpose  of  obtaining  a  correct  plat. 
In  Fig.  25,  let  ABCD,  and  BCEF, 
be  two  square  lots  of  ground,  platted 
horizontally.  Suppose  the  ground  to 
slope  in  all  directions  from  the  point 
C,  which  is  the  summit  of  a  hUl. 
Then  the  lines  BC,  DC,  measured  on 
the  slope,  are  longer  than  if  measur- 
ed on  a  level,  and  the  field  ABCD, 
of  Fig.  25,  platted  with  these  long 
lines,  would  take  the  shape  ABGD 
in  Fig.  26 ;  and  the  field  BCEF, 
of  Fig.  25,  would  become  BHEF  of 
Fig.  26.  The  two  adjoining  fields  would  thus  overlap  each  other ; 
and  the  same  difficulty  would  occur  in  every  case  of  platting  anj 
two  adjoimng  fields  by  the  measurements  made  on  tlie  slope. 


CHAP     IV.] 


Calculating  the  Content. 


39 


Let  us  suppose  another  case,  fis-  27 

more  simple  than  would  ever  oc- 
cur in  practice,  that  of  a  three- 
sided  field,  of  equal  sides  and 
composed  of  three  portions  each 
sloping  down  uniformly,  (at  the 
rate  of  one  to  one)  from  one  point  in  the  centre,  as  in  Fig,  27. 
Each  slope  being  accurately  platted,  the  three  could  not  come 
together,  but  would  be  separated  as  in  Fig.  28. 

We  have  here  taken  the  most  simple  cases,  those  of  uniform 
slopes.  But  with  the  common  irregularities  of  uneven  ground,  to 
measui'e  its  actual  surface  would  not  only  be  improper,  but  impos- 
sible. 

In  the  Social  aspect  of  tliis'question,  the  horizontal  measurement 
is  justified  by  the  fact  that;no  more  houses  can  be  built  on  a  hill 
than  could  be  bmlt  on  its  flat  base  ;  and  that  no  more  trees,  corn, 
or  other  plants,  which  shoot  up  vertically,  can  grow  on  it;  as  is 
represented  by  the  vertical  lines  in  the  FJ--  29- 

Figure.*  Even  if  a  side  hill  should  pro- 
duce more  of  certain  creeping  plants,  the  j^ul 
increased  diflSculty  in  their  cultivation. might  perhaps  balance  this. 
For  this  reason  the  surface  of  the  soil  thus  measured  is  sometimes 
called  the  ijroductive  base  of  the  ground. 

Again,  a  piece  of  land  containing  a  hill  and  a  hollow,  if  measured 
on  the  surface  would  give  a  larger  content  than  it  would  after  the 
hollow  had  been  filled  up  by  the  hill,  while  it  would  yet  really  be 
of  greate  r  value  than  before. 

Horizontal  measurement  is  called  the  "  Method  of  Cultellation," 
and  Superficial  measui-ement,  the  "  Method  of  Developement."t 

An  act  of  the  State  of  New-York  prescribes  that  "  The  acre,  for 
land  measure,  shall  be  measured  horizontally," 

*  This  question  is  more  than  two  thousand  years  old,  for  Polybius  writes, 
'Some  even  of  those  wlio  are  employed  in  the  administration  of  states,  or  placed 
at  the  head  of  armies,  imagine  that  unequal  and  hilly  ground  will  contain  more 
houses  than  a  surface  which  is  flat  and  level.  This,  however,  is  not  the  truth. 
For  the  houses  being  raised  in  a  vertical  line,  form  right  angles,  not  with  the  de- 
clivity of  the  ground,  but  with  the  flat  surface  which  lies  below,  and  upon  which 
the  hills  themselves  also  stand." 

t  The  former  from  CvUellvm,  a  knife,  as  if  the  hills  were  sliced  off;  the  laltei 
•o  named  because  it  strips  off  or  unfolds,  as  it  were,  the  surface. 


-r 


40  FUNDAMENTAL  OPERATIONS.  [paet  i 

(57)  Unit  of  Content.  The  Ac?e  is  the  unit  of  land-measure- 
ment. It  contaiiis  4  Roods.  A  Mood  contains  40  Perches.  A 
Percli  is  a  square  Rod ;  otherwise  called  a  Perch,  or  Pole.  A 
Rod  is  6^  yards,  or  16^  feet. 

Hence,  1  acre  =  4  Roods  =  160  Perches  =  4,840  square 
yards  =  43,560  square  feet. 

One  square  mile  =  5280  X  5280  feet  =  640  acres. 

Since  a  chain  is  QQ  feet  long,  a  square  chain  contains  4356 
square  feet ;  and  consequently  ten  square  chains  make  one  acre* 

In  different  parts  of  England,  the  acre  varies  greatly.  The 
statute  acre,  as  in  the  United  States,  contains  160  square  perches 
of  16|  feet,  or  43,560  square  feet.  The  acre  of  Devonshire  and 
Somersetshire,  contains  160  perches  of  15  feet,  or  36,000  square 
feet.  The  acre  of  Cornwall  is  160  perches  of  18  feet,  or  51,840 
square  feet.  The  acre  of  Lancashire  is  160  perches  of  21  feet,  or 
70,560  square  feet.  The  acre  of  Cheshire  and  Staffordshu'e,  is 
160  perches  of  24  feet,  or  92,160  square  feet.  The  acre  of  Wilt- 
shire is  120  perches  of  16^  feet,  or  32,670  square  feet.  The  acre 
in  Scotland  consists  of  10  square  chams,  each  of  74  feet,  and  there- 
fore contains  54,760  square  feet.  The  acre  in  Ireland  is  the 
same  as  the  Lancashire.     The  chain  is  84  feet  long. 

The  French  units  of  land-measure  are  the  Are  =  100  square 
Metres,  =Q.Q24l1  acre,  =  one  fortieth  of  an  acre,  nearly ;  and  the 
Hectare  =  100  Ares  =  2.47  acres,  or  nearly  two  and  a  half. 
Their  old  land-measures  were  the  "  Arpent  of  Paris,"  eontaining 
36,800  square  feet;  and  the  "Arpent  of  Waters  and  Woods," 
containing  55,000  square  feet. 

(58)  When  the  content  of  a  piece  of  land  (obtained  by  any  of 
the  methods  to  be  explained  presently)  is  given  in  square  links,  as 
is  customary,  cut  off  four  figures  on  the  right,  (i.  e.  divide  by 
10,000),  to  get  it  mto  square  chains  and  decimal  parts  of  a  chain  ; 
cut  off  the  right  hand  figure  of  the  square  chains,  and  the  remain- 
ing figures  will  be  Ao'es.  Multiply  the  remainder  by  4,  and  the 
figure,  if  any,  outside  of  the  new  decimal  point  will  be  Moods. 

*  Let  tlip  yoriig  stiulenl  hewfire  of  confounding  IC  sqnare  chains  with  JO 
chains  sciuare.    The  former  make  one  acve ;  the  latter  s)>ace  contains  ten  acres 


CHAP.    IV. J 


Calculatinsr  the  Content. 


41 


Multiply  the  remainder  by  40,  and  the  outside  figures  will  be 
Perches.     The  nearest  round  number  is  usually  taken  for  the 
Peiches ;  fractions  less  than  a  half  perch  being  disregarded.* 
Thus,  86.22  square  chams=  8  Acres   2  Roods   20  Perches. 


Also, 

64.1818 

do. 

=  6  A. 

1  R. 

27  P. 

43.7564 

do. 

=  4  A. 

IR. 

20  P. 

71.1055 

do. 

=  7  A. 

OR. 

18  P. 

82.50 

do. 

=  8  A. 

IR. 

OP. 

8.250 

do. 

=  0  A. 

3R. 

12  P. 

0.8250 

do. 

=  0  A. 

OR. 

13  P. 

(59)  The  follo^-ing  Table  gives  by  mere  inspection  the  Rooda 
and  Perches  corresponding  to  the  Decimal  parts  of  an  Acre.  It 
explains  itself. 


ROODS. 

Perches. 

— 

ROODS. 

Perches. 

0 

1 

2 

3 

0 

1 

2 

3 

.000 

.250 

.500 

.750 

+  0 

.131 

.381 

.631 

.881 

+21 

.006 

.256 

.506 

.756 

+  1 

.137 

.387 

.637 

.887 

+22 

.012 

.262 

.512 

.762 

+  2 

.144 

.394 

.644 

.894 

+23 

.019 

.269 

.519 

.769 

+  3 

.150 

.400 

.650 

.900 

+  24 

.025 

.275 

.525 

.775 

+  4 

.156 

.406 

.656 

.906 

+25 

.031 

.281 

.531 

.781 

4-  5 

.162 

.412 

.662 

.912 

+  26 

i5 

.037 

.287 

.537 

.787 

+  6 

h 

.169 

.419 

.669 

.919 

+27 

.044 

.294 

.544 

.794 

+  7 

ffi 

.175 

.425 

.675 

.925 

+  28 

«*- 

.050 

.300 

.550 

.800 

+  8 

5 

.181 

.431 

.681 

.931 

+  29 

c 

.050 

.306 

.556 

.806 

+  9 

'c 

.187 

.437 

.687 

.937 

+  30 

~ 

.062 

.312 

.562 

.812 

+  10 

ti 

.194 

.444 

.694 

.944 

+  31 

— 

.069 

.319 

.569 

.819 

+  11 

^ 

.200 

.450 

.700 

.950 

+  32 

p 

.076 

.325 

.575 

.825 

+  12 

j: 

.206 

.456 

.706 

.956 

+  33 

"5 

.081 

.331 

.581 

.831 

+  13 

~ 

.212 

.462 

.712 

.962 

+  84 

C 

.087 

.337 

.587 

.837 

+  14 

C. 

^219 

.469 

.719 

.969 

+  35 

.094 

.344 

.594 

.844 

+  15 

.225 

.475 

.725 

.975 

+  36 

.100 

.350 

.600 

.850 

+  16 

.231 

.481 

.731 

.981 

+37 

.106 

.356 

.606 

.856 

+  17 

.237 

.487 

.737 

.987 

+  38 

.112 

.362 

.612 

.862 

+  18 

.244 

.494 

.744 

.994 

+  39 

.119 

.369 

.619 

.869 

+  19 

.250 

.500 

.750 

1.000 

+40 

.125  .375 

.625 

.875 

-J- 20 

(60)   Chain  Correction.     ^Mien  a  survey  has  been  made,  and 
the  plat  has  been  dra\ni,  and  the  content  calculated ;  and  after* 


•  To  reduce  sfjiinre  yards  to  acre?,  instead  of  dividing  by  4840,  it  is,  easier,  and 
Tery  nearly  correct,  to  multiply  by  2,  cut  off  four  figures,  and  add  to  this  product 
wnothird  of  one-tenth  of  itself. 


42  FUXDAMEATAL  OPERATIO\S.  [part  l 

wards  the  chain  is  found  to  have  been  incorrect,  too  short  or  too 
long,  the  true  content  of  the  land,  may  be  found  by  this  proportion : 
As  the  square  of  the  length  of  the  standard  given  by  the  incorrect 
chain  Is  to  the  square  of  the  true  length  of  the  standard  So  is  the  cal- 
culated content  To  the  true  content.  Thus,  suppose  that  the  chain 
used  had  been  so  stretched  that  the  standard  distance  measured  by 
it  appeal's  to  be  only  99  links  long ;  and  that  a  square  field  had 
been  measured  by  it,  each  side  containing  10  of  these  long  chains, 
and  that  it  had  been  so  platted.  This  plat,  and  therefore  the  con- 
tent calculated  from  it,  will  be  smaller  than  it  should  be,  and  the 
correct  content  will  be  found  by  the  proportion  99^  :  1002  ;  :  IQO 
sq.  chains  :  102.03  square  chains.  If  the  chain  had  been 
stretched  so  as  to  be  101  ti'ue  links  long,  as  found  by  comparing 
it  with  a  correct  chain,  the  content  would  be  given  by  this  propor- 
tion: 100^  :  101^  : ;  100  square  chains  :  102.01  square  chains. 
In  the  former  case,  the  elongation  of  the  chain  was  l^^g  true  links ; 
and  1002  .  (101 J^)^  : :  100  square  chains  :  102.03  square 
chains. 

(61)  Boundary  Lines.  The  lines  wliich  are  to  be  considered 
as  boimding  the  land  to  be  surveyed,  are  often  very  uncertaia, 
unless  specified  by  the  title  deeds. 

K  the  boundary  be  a  brook,  the  middle  of  it  is  usually  the  boun- 
dary line.  On  tide-waters,  the  land  is  usually  considered  to  extend 
to  low  water  mark. 

Wliere  hedges  and  ditches  are  the  boundaries  of  fields,  as  is 
almost  universally  the  case  in  England,  the  dividing  line  is  gene- 
rally the  top  edge  of  the  ditch  farthest  from  the  hedge,  both  hedge 
and  ditch  belonging  to  the  field  on  the  hedge  side.  This  varies, 
however,  with  the  customs  of  the  locality.  From  three  to  six  feet 
from  the  roots  of  the  quickwood  of  the  hedges  are  allowed  for  the 
ditches 


CHAP.  iv.J  Calculating  the  Content.  43 

METHODS    OF    CALCULATION. 

(62)  The  various  methods  employed  in  calculating  the  content 
of  a  piece  of  ground,  may  be  reduced  to  foui-,  which  may  be  called 
Arithmetical,  Gf-eometrical,  Instrumental,  and  Trigonometrical. 

(63)  FIRST  METHOD.— ARITHMETICALLY.  From  dire:^ 
measurements  of  the  necessary  lines  on  the  ground. 

The  figures  to  be  calculated  by  this  method  may  be  either  the 
Bhapes  of  the  fields  which  are  measured,  or  those  into  which  the 
fields  can  be  divided  by  measuring  various  lines  across  them. 

The  familiar  rules  of  mensuration  for  the  principal  figures  which 
occur  in  practice,  will  be  now  briefly  enunciated. 

(64)  Rectang^les.  If  the  piece  of  ground  be  rectangular  in 
shape,  its  content  is  found  by  multiplying  its  leng-th  by  its  breadth. 

(65)  Triangles.  When  th^^ven  quantities  are  one  side  of  a 
triangle  and  the  perpendicular  distance  to  it  from  the  opposite 
angle ;  the  content  of  the  triangle  is  equal  to  half  the  product  of 
the  side  and  the  perpendicular. 

When  the  given  quantities  are  the  three  sides  of  the  ti-iangle ; 
add  together  the  three  sides  and  divide  the  sum  by  2 ;  from  this 
half  sum  subtract  each  of  the  three  sides  in  turn ;  multiply  together 
the  half  sum  and  the  three  remainders  ;  take  the  square  root  of  the 
product ;  it  is  the  content  required.  If  the  sides  of  the  triangle 
be  designated  by  a,  h,  c,  and  their  sum  by  s,  this  rule  will  give  its 
area=  ^^s  (^s  — a)  (hj  —  h)  Qs  — 0]-*      • 


*  Wlien  two  sides  of  a  tiiangle,  and  the  incluil"il 
angle  are  given,  its  content  equals  half  the  prodnct 
cf  its  sides  into  the  sine  of  the  included  angle.  D<^- 
signating  the  angles  of  the  ti-iangle  by  the  capital 
letters  A,B,C,  and  the  sides  opposite  them  by  the  cor- 
responding small  letters  a,b,c,  the  area  =  J  ic  sin.  A.         

When  one  side  of  a  triangle  and  the  adjacent  an-    "'*'  P 

gles  are  given,  its  content  equals  the  square  of  the  gi^-pn  side  multiplied  by  the 
iines  of  each  of  the  given  angles,  and  divided  by  twice  the  sine  of  the  sum  of 
,  111..  ^"-  B .  ein,  C 

these  angles.     Using  the  same  symbols  as  before,  the  area  =^2    ogin  (bVc) 

When  the  three  angles  of  a  triangle  and  its  altitude  are  given,  its  area,  referring 

sin.  B      T'\  , 
to  the  above  figure,  =  i  BD2  .  -^-^-—^....^  , ,  ^.  -^^  ^^  g  ^  ,  ^ 


44  FUNDAMENTAL  OPERATIONS.  [part  i 

(66)  Parallelosjraras ;  or  four-sided  figures  whose  opposite 
Bides  are  parallel.  The  content  of  a  Parallelogram  equals  the 
product  of  one  of  its  sides  hj  the  perpendicular  distance  between  it 
and  the  side  parallel  to  it. 

(67)  Trapezoids;  or  four-sided  figui^s,  tivo  opposite  sides  of 
which  are  parallel.  The  content  of  a  Trapezoid  equals  half  the 
product  of  the  sum  of  the  parallel  sides  bj  the  perpendicular  dis- 
tance between  them. 

If  the  given  quantities  are  the  four  sides  a,  h,  c,  d,  of  which  b 
and  d  are  parallel ;  then,  making  q  =  ^  (a  +  b  +  c  —  d),  the  area 

of  the  trapezoid  will  =  ^  /  [q  {q  —  a)  (q — c)  (q — b  -\-  cZ).]* 

0  —  a  ....  -    .,i-c,xa 

(68)  Quadrilaterals,  or  Trapeziums ;  four-sided  figures,  none 
of  whose  sides  are  parallel. 

A  very  gross  error,  often  committed  as  to  this  figure,  is  to  take 
the  average,  or  half  sum  of  its  opposite  sides,  and  multiply  them 
together  for  the  area :  thus,  assuming  the  trapezium  to  be  equiva- 
lent to  a  rectangle  with  these  averages  for  sides. 

In  practical  surveying,  it  is  usual  to  measure  a  hue  across  it 
from  corner  to  corner,  thus  dividing  it  into  two  triangles,  whose 
sides  are  known,  and  which  can  therefore  be  calculated  by  Art.  (65). f 

*  When  two  parallel  sides,  b  and  d,  and  a  third  side,  a,  are  given,  and  also  the 
angle,  C,  which  this  third  side  makes  with  one  of  the  parallel  sides,  then  the 
content  of  the  trapezoid= a  .  sin.  0.         ^^ 

t   When  two  opposite  sides,  and  all  the  angles  are  given,  take  one  side  and  its  ad* 
jacent  angles,  (or  their  snpplements,  when  their  sum  exceeds  180°),  conside'' 
them  as  belonging  to  a  triangle,  and  find  its  area  by  the  second  formula  in  tbt    -/' 
note  on  page  43.     Do  the  same  with  the  other  side  and  its  adjacent  angles.     The    -_ — ■ 
difference  of  the  two  areas  will  be  the  area  of  the  quadrilateral. 

When  three  sides  and  their  two  included  angles  are  given,  multiply  together  the  sin<3 
of  one  given  angle  and  its  adjacent  sides.  Do  the  same  with  the  sine  of  the  other 
given  angle  and  its  adjacent  sides.  Multiply  together  the  two  opposite  sides  and 
the  sine  of  the  supplement  of  the  sum  of  the  given  angles.  Add  together  the  first 
'two  products,  and  add  also  the  last  product,  if  the  sum  of  the  given  angles  ia 
more  than  180°  or  subtract  it  if  this  sura  be  less,  and  take  half  the  result.  Call- 
ing the  given  sioes,  p,  q,  r  ;  and  the  angle  between  p  and  ^  =  A  ;  and  the  angle 
between  q  and  r  =  B  ;  the  area  of  the  quadrilateral 

=  hip  -q  si»-  A.  -^  q.r.  sin.  B  ±  p  .r  sin.  ("180°  —  A  —  B)]. 
When  the  four  sides  and  the  sum  of  any  two  opposite  angles  are  given,  proceed 
thus :  Take  half  the  sum  of  the  four  given  sides,  and  from  it  subtract  each  side 
in  turn  Multiply  together  the  four  remainders,  and  reserve  the  product.  Mul- 
tiply together  tlie  four  sides.  Take  half  their  product,  and  multiply  it  by  thfl 
tjaine  of  tne  given  sum  of  the  angles  increased  bj  unity.     Regard  the  sign  of 


CHAP.    IV.] 


falculatinsr  the  fontcni, 


45 


(69)  Surfaces  hounded  hy  irregularly  curved  lines.  The  rules 
for  these  will  be  more  appropriately  given  in  connection  with  the 
surveys  which  measure  the  necessary  lines ;  as  ex^Dlained  in  Part 
II,  Chap.  III. 


(70)  SECOND  METHOD.— GEOMEIRU ALLY.  From  mea- 
surements of  the  necessary  lines  upon  the  plat. 

(71)  Division  into  Triangles.  The  plat  of  a  piece  of  ground 
having  been  dra-\yn  from  the  measurements  made  by  any  of  the 
methods  which  will  be  hereafter  explained,  lines  may  be  drawn 
upon  the  plat  so  as  to  divide  it  into  a  number  of  triangles.     Four 

Fig.  31.  Fig.  32.  Fig.  33.  Fig.  34. 


ways  of  doing  this  are  shown  in  the  figures :  viz.  by  drawing  lines 
from  one  corner  to  the  other  corners ;  from  a  point  in  one  of  the 
sides  to  the  corners ;  from  a  point  inside  of  the  figure  to  the  cor- 
ners ;  and  from  various  comers  to  other  comers.  The  last  method 
IS  usually  the  best.  The  fines  ought  to  be  drawn  so  as  to  make 
the  triangles  as  nearly  equilateral  as  possible,  for  the  reasons  given 
in  Part  V. 

One  side  of  each  of  these  triangles,  and  the  length  of  the  per- 
pendicular let  fall  upon  it,  being  then  measured,  as  directed  in 
Art.  (43,)  the  content  of  these  triangles  can  be  at  once  obtained 
by   multiplying  their  base  by  their  altitude,  and  dividing  by  two. 

The  easiest  method  of  getting  the  length  of  the  perpendicular, 
without  actually  drawing  it,  is,  to  set  one  point  of  the  dividers 
at  the  angle  from   which  a  perpendicular  is  to  be  let  fall,  and  to 

the  cosine.  Subtract  this  product  from  the  reserved  product,  and  take  the  square 
root  of  the  remainder.     It  will  be  the  area  of  the  quadrilateral. 

When  the  four  sides,  and  the  angle  of  intersection  of  the  diagonals  of  the  qvadrHa. 
leral  are  given;  square  each  side;  add  together  the  squares  of  the  opposite 
sides ;  take  the  difference  of  the  two  sums ;  multiply  it  by  the  tangent  of  the 
angle  of  intersection,  and  divide  by  four.     Tlje  quotient  will  be  the  area. 

When  the  diagonals  of  the  quadrilateral,  and  their  inclnded  angle  are  given,  mul« 
tiply  together  the  two  diagonals  and  tlie  sine  of  their  included  angle,  aud 
divide  by  two      The  quotient  will  be  the  aren 


46  FlfWMWEKTAL  OPERATIOINS.  [part  i 

open  and  shut  their  legs  till  an  arc  described  bj  the  other  point 
will  just  touch  the  opposite  side. 

Otherwise ;  a  platting  scale,  (described  in  Art.  (49)  may  bo 
placed  so  that  the  zero  point  of  its  edge  coincides  with  the  angle, 
and  one  of  its  cross  lines  coincides  with  the  side  to  which  a  perpen- 
dicular is  to  be  drawn.  The  length  of  the  perpendicular  can  then 
at  once  be  read  oiF. 

The  method  of  dividing  the  plat  into  triangles  is  the  one  most 
commonly  employed  by  surveyors  for  obtaining  the  content  of  a 
survey,  because  of  the  simplicity  of  the  calculations  required.  Its 
correctness,  however,  is  dependant  on  the  accuracy  of  the  plat, 
and  on  its  scale,  which  should  be  as  large  as  possible.  Three 
chains  to  an  inch  is  the  smallest  scale  allowed  by  the  English 
Tithe  Commissioners  for  plats  from  which  the  content  is  to  be 
determined. 

In  calculating  in  this  way  the  content  of  a  farm,  and  also  of  its 
separate  fields,  the  sum  of  the  latter  ought  to  equal  the  former. 
A  diflference  of  one  three-hundredth  (3^0)  is  considered  allowable. 

Some  surveyors  measure  the  perpendiculars  of  the  triangles  by 
a  scale  half  of  that  to  which  the  plat  is  made.  Thus,  if  the  scale 
of  the  plat  be  2  chains  to  the  inch,  the  perpendiculars  are  mea- 
sured with  a  scale  of  one  chain  to  the  inch.  The  product  of  the 
base  by  the  perpendicular  thus  measured,  gives  the  area  of  the 
triangle  at  once,  without  its  requirmg  to  be  divided  by  two. 

Another  way  of  attaining  the  same  end,  with  less  danger  of  mia- 
takes,  is,  to  construct  a  new  scale  of  equal  parts,  longer  than  those 
by  which  the  plat  was  made  in  the  ratio  \/2;l;  or  1.414:1. 
When  the  base  and  perpendicular  of  a  triangle  are  measured  by 
this  new  scale  and  then  multiphed  together,  the  product  will  be 
the  content  of  the  triangle,  without  any  division  by  two.  In  this 
method  there  is  the  additional  advantage  of  the  greater  size  and 
consequent  greater  distinctness  of  the  scale. 

When  the  measurement  of  a  plat  is  made  some  time  after  it  has 
been  dra-wn,  the  paper  \d\\  very  probably  have  contracted  oi 
expanded  so  that  the  scale  used  will  not  exactly  apply.  In  that 
case  a  correction  is  necessary.  Measure  very  precisely  the  present 
length  of  some  line  m  the  plat,  of  known  length  originally.     Then 


CHAP.  IV.] 


Calciilatin&r  Ihc  Content. 


47 


make  this  proportion :  As  the  square  of  the  present  length  of  this 
line  Is  to  the  square  of  its  original  length,  So  is  the  content  obtain 
ed  by  the  present  measurement  To  the  true  content. 

,  (72)  Ciraplacal.  Multiplication.  Prepare  a  strip  of  drawing 
V*  paper,  of  a  width  exactly  equal  to  two  chains  on  the  scale  of  the 
plat ;  i.  e.  one  inch  wde,  as  in  the  figure,  for  a  scale  of  two  chains 
to  1  mch  ;  two-thirds  of  an  inch  wide  for  a  scale  of  3  chains  ;  half 
an  inch  for  4  chains  ;  and  so  on.  Draw  perpendicular  lines  across 
the  paper  at  distances  representing  one-tenth  of  a  chain  on  the  scale 
of  the  triangle  to  be  measured,  thus  making  a  platting  scale.  Apply 
it  to  the  triangle  so  that  one  edge  of  the  scale  shall  pass  through 
one  comer,  A,  of  the  triangle,  and  the  other  edge  through  another 

Fiff.  35. 


corner,  B  ;  and  note  very  precisely  what  divisions  of  the  scale  are 
at  these  points.  Then  slide  the  scale  in  such  a  way  that  the 
points  of  the  scale  which  had  coincided  with  A  and  B,  shall  always 
remain  on  the  line  BA  produced,  till  the  edge  arrives  at  the  point 
C.  Then  will  A'C,  that  is,  the  distance,  or  number  of  divisions  on 
the  scale,  from  the  point  to  which  the  division  A  on  the  scale  has 
arrived,  to  the  third  corner  of  the  triangle,  express  the  area  of  the 
triangle  ABC  in  square  chains.* 

"For,  from  C  draw  a  parallel  to  AB,  meeting  the  edge  of  die  scale  in  C'i  and 
draw  C'B.  Then  the  given  triangle  ABC  =  ABC  But  the  area  of  this  last 
triangle  =  AC  multiplied  by  half  the  width  of  the  scale,  i.  e.  =  AC  X  1  =  AC. 
But,  because  of  the  parallels,  A'C  =  AC.  Therefore  the  area  of  the  given  trian- 
gle ABC  =  A'C  i.  e.  it  is  equal  in  square  chains  to  the  number  of  linear  chaini 
read  off  from  the  scale.    This  ingenious  operation  is  due  to  M.  Cousincry. 


48 


FUNDAMEOTAL  OPERATIONS. 


[part  I. 


(73)  DivisioE  into  Trapezoids.     A  line  may  be  dravni  across 


Fig.  36. 


the  field,  as  iii  Fig.  36,  and  perpen- 
diculars drawn  to  it.  The  field  will 
thus  be  divided  iato  trapezoids,  (ex- 
ceptuig  a  triangle  at  each  end), 
and  their  content  can  be  calculated 
by  Art.  (67). 

Otherwise  ;  a  line  may  be  drawn 
outside  of  the  figure,  and  per- 
pendiculars to  it  be  drawn  from 
each  angle.  In  that  case  the 
difierence  between  the  trapezoids 
formed  by  lines  drawn  to  the 
outer  angles  of  the  figure,  and 
those  drawn  to  the  imier  angles, 
will  be  the  content. 

This  method  is  very  advantageously  applied  to  surveys  by  the 
compass ;  as  will  be  explamed  in  Part  III,  Chap.  VI. 


(74)  Division  into  Squares.  Two  sets  of  parallel  lines,  at 
right  angles  to  each  other,  y-     gg 

one  chain  apart  (to  the  scale 
of  the  plat)  may  be  drawn 
over  the  plat,  so  as  to  divide 
it  into  squares,  as  in  the 
figure.  The  number  of 
squares  which  fall  within  the 
plat  represent  so  many  square 
chains ;  and  the  triangles  and 
trapezoids  which  fall  outside 
of  these,  may  then  be  calcu- 
lated and  added  to  the  entire  square  chains  which  have  been 
counted. 

Instead  of  drawing  the  parallel  lines  on  the  plat,  they  may  bet- 
ter be  drawn  on  a  piece  of  transparent  "  tracing  paper,"  which  ia 
simply  laid  upon  the  plat,  and  the  squares  counted  as  before.     The 


OHAP.  IV.]  Calculating  the  Content.  49 

same  paper  will"  ianswer  for  any  number  of  plats  drawn  to  the  same 
scale.  This  method  is  a  valuable  and  easy  check  on  the  results  of 
other  calculations. 

To  calculate  the  fractional  parts,  prepare  a  piece .  of  tracing 
paper,  or  horn,  by  drawing  on  it  one  square  of  the  same  size  as  a 
square  of  the  plat,  and  subdividuag  it,  by  two  sets  of  ten  parallels  at 
right  angles  to  each  other,  into  hundredths.  This  wiU  measure  the 
fractions  remaining  from  the  former  measurement,  as  nearly  as  caa 
be  desired. 

(75)  Division  into  Parallelograms.  Draw  a  series  of  paral- 
lel lines  across  the  plat  at  equal  distances  depending  on  the  scale. 
Thus,  for  a  plat  made  to  a  scale  of  2  chains  to  1  inch,  ihe  distance 
between  the  parallels  should  be  2|  inches ;  for  a  scale  of  3  chains 
to  1  inch,  IJ  inch ;  for  a  scale  of  4  chains  to  1  inch,  |  inch ;  for 
a  scale  of  5  chains  to  1  inch,  y%  inch  ;  and  for  any  scale,  make  the 
distance  between  the  parallels  that  fraction  of  an  inch  which  would 
be  expressed  by  10  divided  by  the  square  of  the  number  of  chama 
to  the  inch.  Then  apply  a  common  inch  scale,  divided  on  the 
edge  into  tenths,  to  these  parallels ;  and  every  inch  in  length  of 
the  spaces  included  between  each  pair  of  them  will  be  an  acre,  and 
every  tenth  of  an  mch  will  be  a  square  chain.* 

To  measure  the  triangles  at  the  ends  of  the  strips  between  the ' 
parallels,  prepare  a  piece  of  transparent  horn,  or  stout  tracing 
paper,  of  a  width  equal  to  the  width  between  the  parallels,  and 
draw  a  line  through  its  middle  longitudinally.     Apply  it  to  the 

oblique  line  at  the  end  of  the  space  between    Fig.  39. 

two  parallels,  and  it  will  bisect  the  line,  and 
thus  reduce  the  triangle  to  an  equivalent 
rectangle,  as  at  A  in  the  figure.  When  an 
angle  occurs  between  two  parallels,  as  at  B 
in  the  figure,  the  fractional  part  may  be 
measured  by  any  of  the  preceding  methods. 

"  For,  calling  the  number  of  chains  to  the  inch,  =  w,  and  making  the  width  be 
tween  the  parallels  —  inch,  this  width  will  represent  —  X  n  =  — .chains ;  and 

M  the'inch  length  represents  «  chains,  their  product,  —  X  n=10  square  chaini 

n 
"  1  acre. 


50 


FUNDAMENTAL  OPERATIONS. 


[part  1 


A  somewhat  similar  method  is  much  used  by  some  surveyors, 
particularly  in  Ireland :  the  plat  being  made  on  a  scale  of  5  chains 
to  1  inch,  parallel  hnes  being  drawn  on  it,  half  an  inch  apart,  and 
the  distances  along  the  parallels  beuag  measured  by  a  scale,  each 
large  division  of  which  is  j^  mch  in  length.  Each  division  of  this 
scale  indicates  an  acre  ;  for  it  represents  4  chains,  and  the  distance 
between  the  parallels  is  2-|  chains.  This  scale  is  called  the  "  Scale 
of  Acres." 


(76)  Addition  of  Widths.     When  the  Imes  of  the  plat  are  veiy 
irregularly  curved,  as  in  the  Fig.  40. 

figure,  draw  across  it  a  num- 
ber of  equi-distant  lines  as  near 
together  as  the  case  may  seem 
to  requii'e.  Take  a  straight- 
edged  piece  of  paper,  and  apply  one  edge  of  it  to  the  middle  of 
the  first  space,  and  mark  its  length  from  one  end ;  apply  the  same 
edge  to  the  middle  of  the  next  space,  bringing  the  mark  just  made 
to  one  end,  and  making  another  mark  at  the  end  of  the  additional 
length ;  so  go  on,  adding  the  length  of  each  space  to  the  previous 
or«s.  "When  all  have  been  thus  measured,  the  total  length,  mul- 
tiplied by  the  uniform  width,  will  give  the  content. 

(77)  THIRD  METHOD.— INSTRUMENTALLY.     By  perform- 
ing  certain  instrumental  operations  on  the  plat. 

(78)  Reduction  of  a  many  sided  figure  to  a  single  equivalent 
triangle.  Any  plane  figure  bounded  by  straight  lines  may  be 
reduced  to  a  single  triangle,  which  shall  have  the  same  content. 
This  can  be  done  by  any  instrument  for  drawing  parallel  lines, 
such  as  those  described  in  Art. 
(39).  Let  the  trapezium,  or 
four  sided  figure,  shown  in  Fig. 
41,  be  requii'ed  to  be  reduced 
to  a  single  equivalent  triangle. 
Produce  one  side  of  the  figure, 

as  4 1.     Draw  a  line  from 

the  first  to  the  third  angle  of 


Fig.  41 


CHAP.  IV.] 


Calculaiiiii?  the  Content 


51 


the  figure.  From  the  second  angle  draw  a  parallel  to  the  line  just 
dra-wn,  cutting  the  produced  side  in  a  point  1'.  From  the  point  1' 
draw  a  line  to  the  third  angle.  A  triangle  (1'  —  3  —  4  in  the 
figure)  will  thus  be  formed,  which  will  be  equivalent  to  the  original 
tra^eziiun.* 

The  content  of  this  final  triangle  can  then  be  foimd  by  measur- 
ing its  perpendicular,  and  taking  half  the  product  of  this  perpendi- 
cular by  the  base,  as  in  the  first  paragraph  of  Art.  (65). 

(79)  Let  the  given  figui-e  have  five  sides,  as  in  Fig.  42.  For 
brevity,  the  angles 
of  the  figure  wiU  be 
named  as  numbered 
in  the  engraving. 
Produce  5 — 1. 
Join  1  —  3.  From 
2  draw  a  parallel  to 

1  —  3,  cutting  the    Z'~  T       T 

produced  base  in  1'.  Join  1'  —  4.  From  3  draw  a  parallel  to  it, 
cutting  the  base  in  2'.  Join  2'  —  4.  Then  will  the  triangle 
2' — 4 — 5  be  equivalent  to  the  five  sided  figure  1 — 2 — 3  — 4 — 5 , 
for  similar  reasons  to  those  of  the  preceding  case. 


(80)  Let  the  given  figure  be  1  —  2  —  3—4—5  —  6  —  7  —  8, 
as  shown  in  Fig.  43,  given  at  the  top  of  the  following  page.  All 
the  operations  are  shown  by  dotted  lines,  and  the  finally  resulting 
triangle  6' — 7  —  8,  is  equivalent  to  the  original  figure  of  eight 
sides. 

It  is  best,  in  choosing  the  side  to  be  produced,  to  take  one  which 
has  a  long  side  adjoining  it  on  the  end  not  produced  ;  so  that  this 
long  side  may  form  one  side  of  the  final  triangle,  the  base  of  which 
will  therefore  be  shorter,  and  wUl  not  be  cut  so  acutely  by  the 
final  line  drawn,  as  to  make  the  point  of  intersection  too  indefinite. 


*  Far,  the  triangle  1 — 2 — 3  taken  away  from  the  original  figure  is  equivalent 
1  the  triangle  1' — 1 — 3  added  to  it  ;  because  both  these  triangles  have  the  same 


base  and  also  the  same  altitude,  since  the  vertices  of  both  lie   in   tht 
parallel  tu  the  base. 


same  line 


-+ 


52 


FUNDA3IEi\TAL  OPERATIONS. 


Fig.  43 


[part  I 


(81)  General  Huh.  When  the  given  figure  has  many  sides^ 
with  angles  sometimes  salient  and  sometimes  re-entering,  the  opera- 
tions of  reduction  are  very  liable  to  errors,  if  the  draftsman  attempt? 
to  reason  out  each  step.  AU  difficulties,  however,  will  be  removeu 
by  the  foUowing  G-eneral  Rule  : 

1.  Produce  one  side  of  the  figure,  and  caU  it  a  base.  Call  one 
of  the  angles  at  the  base  the  first  angle,  and  number  the  rest  in 
regular  succession  around  the  figure. 

2.  Draw  a  line  from  the  1st  angle  to  the  3d  angle.  Draw  a 
parallel  to  it  from  the  2d  angle.  Call  the  intersections  of  this 
parallel  with  the  base  the  1st  mark.  * 

3.  Draw  a  line  from  the  1st  mark  to  the  4th  angle.  Draw  a 
parallel  to  it  from  the  3d  angle.  Its  intersection  with  the  base  ia 
the  2d  mark. 

4.  Draw  a  line  from  the  2d  mark  to  the  5th  angle.  Draw  a 
parallel  to  •;  frjm  the  4th  angle.  Its  intersection  with  the  base  ia 
the  3d  mark. 

5.  In  general  terms,  which  apply  to  every  step  after  the  first, 
draw  a  line  from  the  last  mark  obtained  to  the  angle  whose  number 
is  greater  by  three  than  the  number  of  the  mark.  Draw  a  parallel 
to  it  through  the  angle  whose  number  is  greater  by  two  than  that 
of  the  mark.  Its  intersection  with  the  base  will  be  a  mark  whose 
number  is  greater  by  one  than  that  of  the  preceding  mark.* 

In  tlie  concise  language  of  Algebra,  draw  a  line  from  the  wth  mark  to  the 
«+3  angle.  Draw  a  parallel  to  it  tiirough  the  «-j-2  angle,  and  the  intersection 
with  the  base  will  be  the  n-J-l  mark. 


GHAJP.    IV.] 


Calculating  the  Content. 


53 


6.  Repeat  tins  process  for  each  angle,  till  jou  get  a  mark  whose 
number  is  such  that  the  angle  having  a  niunber  gi-eater  bj  three  ia 
the  last  angle  of  the  figure,  i.  e.  the  angle  at  the  other  end  of  the 
base.  Then  join  the  last  mark  to  the  angle  which  precedes  the 
last  angle  in  the  figui-e,  and  the  triangle  thus  formed  will  be  the 
equivalent  triangle  required. 

In  practice  it  is  unnecessary  to  actually  draw  the  lines  joining 
the  successive  angles  and  marks,  but  the  parallel  ruler  is  merely 
laid  on  so  as  to  pass  through  them,  and  the  points  where  the 
parallels  cut  the  base  are  alone  marked. 


(82)  It  is  generally  more  convenient,  for  the  reasons  ^ven  at 
the   end  of  Art.  (80),  to  reduce  Fig.  44. 

half  of  the  figure  on  one  side  and 
half  on  the  other,  as  is  shown  in 
Fig.  44,  which  represents  the  same 
field  as  Fig.  42.  The  equivalent 
triangle  is  here  1' — 3 — 2'. 

When  the  figure  has  many  angles, 
they  should  not  be  numbered  con-    i'      i  52,' 

eecutively  all  the  way  around,  but,  after  the  numbers  have  gone 
around  as  far  as  the  angle  where  it  is  intended  to  have  the  vertex 

10  4 


8'^        g='      5  t^  I  1*  a.' 

of  the  final  triangle,  the  numbers  should  be  continued  from  the 


54  FUXDAMEXTAL  OPERATIONS.  [part  i 

other  angle  of  the  base,  as  is  sho^n  in  Fig.  45.     In  it  only  the 
intersections  are  marked  * 

(83)  It  is  sometimes  more  convenient,  not  to  produce  one  of 
the  sides  of  the  figure,  but  to  draw  at  one  end  of  it,  as  at  the  poini 
1  in  Fig.  46,  an  indefinite  line,  usually  a  perpendicular  to  a  lino 

Fis.  46. 


joining  two  distant  angles  of  the  figure,  and  make  this  line  the  base 
of  the  equivalent  triangle  desired.  The  operation  is  shown  by  the 
dotted  lines  in  the  figure.  The  same  General  Rule  applies  to  it, 
as  to  the  previous  figures. 

(84)  Special  Instruments.  A  variety  of  instruments  have 
been  invented  for  the  purpose  of  determining  areas  rapidly  and 
correctly.  One  of  the  simplest  is  the  "  Computing  Seale,''^  which 
is  on  the  same  principles  as  the  Method  of  Art.  (75).  It  is  repre- 
sented in  Fig.  47,  given  on  the  following  page.  It  consists  of  a 
scale  divided  for  its  whole  length  from  the  zero  point  into 
divisions,  each  representing  2|  chains  to  the  scale  of  the  plat. 
The  scale  carries  a  slider,  which  moves  along  it,  and  has  a 
wu'e  drawn  across  its  centre  at  right  angles  to  the  edges  of  the 
scale.  On  each  side  of  this  wire,  a  portion  of  the  shder  equal 
in  length  to  one  of  the  primary,  or  2^  chain,  divisions  of  the  scale, 
is  laid  off  and  divided  into  40  equal  parts. 

This  instrument  is  used  in  connection  with  a  sheet  of  transpa- 
rent paper,  ruled  with  parallel  lines  at  distances  apart  each  equal 
to  one  chain  on  the  scale  of  the  plat.     It  is  plain,  that  when  the 

*  A  figure  with  ciirvea  boundaries  may  be  reduced  to  a  triangle  in  a  similar 
manner.  Straight  lines  must  be  drawn  about  the  figure,  so  as  to  be  partly  in  it 
and  partly  out,  giving  and  taking  about  equal  quantities,  so  that  the  figure  which 
these  lines  form,  shall  be  about  equivalent  to  the  curved  figure.  This  having 
been  done,  as  will  be  further  developed  in  Art.  (124),  the  equivalent  straighJ 
lined  figure  is  reduced  by  the  above  method. 


CHAP.  IV.] 


Calculating  the  Content. 


56 


instrument  is  laid  on  this  paper,  with  its  edge  on  one  of  the 
parallel  lines,  and  the  shder  is  moved  over  one  of  the  divi- 
sions of  2^  chains,  that  one  rood,  or  a  quarter  of  an  acre, 
has  been  measured  between  two  of  the  parallel  hues  on  the 
paper  (since  10  square  chains  make  one  acre) ;  and  that 
one  of  the  smaller  divisions  measures  ojiejjerch  between 
the  same  parallels.  Four  of  the  larger  divisions  give 
one  acre.  The  scale  is  generally  made  long  enough  to 
measure  at  once  five  acres. 

To  apply  this  to  the  plat  of  a  5eld,  or  farm,  lay  the 
transparent  paper  over  it  in  such  a  position  that  two  of 
the  ruled  lines  shall  touch  two  of  the  exterior  points  of 


Fis  47. 


Fig.  48. 


the  boundaries,  as 

at  A  and  B.     Lay 

the  scale,  with  the 

shde  set  to  zero, 

on  the  paper,  in  a 

direction    parallel 

to  the  ruled  lines, 

and    so  that   the 

wire  of  the  slide 

cuts  the  left  hand 

oblique  line  so  as 

to  make  the  spaces  c  and  d  about  equal.     Hold  the 

scale  firm,  and  move  the  slider  till  the  wire  cuts  the 

right  hand  obUque  line  in  such  a  way  as  to  equalize  the 

spaces   e  and  /.     Without  changuig  the  shde,  move 

the  scale  down  the  width  of  a  space,  and  to  the  left 

hand  end  of  the  next  space  ;  begin  there  again,  and  proceed  as 

before. 

So  go  on,  till  the  whole  length  of  the  scale  is  run  out,  (five  acres 
having  been  measured),  and  then  begin  at  the  right  hand  side  and 
work  backwards  to  the  left,  reading  the  lower  divisions,  which  run 
up  to  10  acres.  By  continuing  this  process,  the  content  of  plats 
of  any  size  can  be  obtained. 

A  still  simpler  substitute  for  this  is  a  scale  similarly  divided,  but 
without  an  attached  shde.     In  place  of  it  there  is  used  a  piece  of 


56  FrXDAMENTAL  OPERATIOXS.  [pabt  l 

horn  having  a  line  drawn  across  it  and  rivetted  to  the  end  of  a 
Bhort  scale  of  box-wood,  divided  like  the  former  shde.  It  is  used 
like  the  former,  except  that  at  starting,  the  zero  of  the  short  scale 
and  not  the  line  on  the  horn  is  made  to  coincide  •\\'ith  the  zero  of 
the  long  scale.  The  slide  is  to  be  held  fast  to  the  instrument  when 
this  is  moved. 

The  Pediometer  is  another  less  simple  instrument  used  for  the 
same  object.     It  measures  any  quadrilateral  directly. 

(85)  Some  very  comphcated  instruments  for  the  same  object 
have  been  devised.  One  of  them,  Sang's  JPlanometer,  detenninea 
the  area  of  any  figure,  by  merely  moving  a  point  around  the  out- 
line of  the  surface.  This  causes  motion  in  a  train  of  wheel  work, 
which  registers  the  algebraic  sum  of  the  product  of  ordinates  to 
every  point  ia  that  perimeter,  by  the  increment  of  their  abscissas, 
and  therefore  measm-es  the  included  space. 

Instruments  of  this  kind  have  been  invented  in  Germany  by     J 
Ernst,  Hansen,  and  Wetli.  >^ 

(86)  A  purely  mechanical  means  of  determining  the  area  ot 
any  sui-face  by  means  of  its  weight,  may  be  placed  here.  The  plat 
is  cut  out  of  paper  and  weighed  by  a  dehcate  balance.  The 
weight  of  a  rectangular  piece  of  the  same  paper  containing  just  one 
acre  is  also  found;  and  the  "Rule  of  Three"  gives  the  content. 
A  modification  of  this  is  to  paste  a  tracmg  of  the  plat  on  thin  sheet 
lead,  cut  out  the  lead  to  the  proper  lines  and  weigh  it. 

(87)  FOURTH  METHOD.— TRIGONOHIETRICALLI.    By  eaU 

culating,  from  the  observed  angles  of  the  boundaries  of  the  piece 
of  ground^the  lengths  of  the  lines  needed  for  calculating  the  content. 
This  method  is  employed  for  surveys  made  with  angular  instru- 
ments, as  the  compass,  &c.,  in  order  to  obtain  the  content  of  the 
land  surveyed,  without  the  necessity  of  previously  making  a  plat, 
thus  avoiding  both  that  trouble  and  the  inaccm-acy  of  any  calcular 
tions  founded  upon  it.  It  is  therefore  the  most  accurate  method ; 
but  wiU  be  more  appropriately  explained  in  Part  III,  Chapter  VI, 
under  the  head  of  "  Compasg  Surveying." 


PART  11. 

CHAIN-SURVEYING  ; 

By  the  First  and  Second  Methods  : 

OR 

DIAGONAL  AND  PERPENOICULAR  SURVEYING. 

(88)  The  chain  alone  is  abundantly  sufficient,  without  the  aid 
of  any  other  instrument,  for  making  an  accurate  survey  of  any 
surface,  whatever  its  shape  or  size,  particularly  in  a  district  tolera- 
bly level  and  clear.  Moreover,  since  a  chain,  or  some  substitute 
for  it,  formed  of  a  rope,  of  leather  drivong  rems,  &c.,  can  be 
obtained  by  any  one  in  the  most  secluded  place,  this  method  of 
Surveying  deserves  more  attention  than  has  usually  been  given  to 
it  in  this  country.  It  -will,  ther  '^o^e,  be  fully  developed  in  the 
following  chapters. 

CHAPTER  I. 
SIRVEYDG  BY  DIAGONALS  : 

OR 

By  the  First  3Iethod. 

(89)  /Surveying  by  Diagonals  is  an  application  of  the  First 
MetJiod  of  deteraiining  the  position  of  a  point,  given  m  Art.  (5,)  to 
which  the  student  should  again  refer.  Each  corner  of  the  field  or 
farm  which  is  to  be  sui'veyed  is  "determined"  by  measuring  ita 
distances  from  two  other  points.  The  field  is  then  "  platted  "- 
by  repeating  tliis  process  on  paper,  for  each  comer,  in  a  contrary 
order,  and  the  "  content "  is  obtained  by  some  of  the  methoda 
explained  m  Chapter  IV  of  Part  I. 


58  CVLkm  SCRFEYIIVG.  [pajit  n 

The  lines  -u-hich  are  measured  in  order  to  determine  the  cor- 
ners of  the  field  are  usually  sides  and  diayonals  of  the  irregular 
polygon  which  is  to  be  surveyed.  They  therefore  divide  it  np 
into  triangles ;  whence  this  mode  of  surveying  is  sometimes  call- 
ed "  Chain  Triangulation." 

A  few  examples  will  make  the  priaciple  and  practice  perfectly 
clear.  Each  ^\ill  be  seen  to  require  the  three  operations  of  measur- 
ing, 'platting,  and  calculating. 

(90)  A  three-sided  field ;  as  Fig.  49. 

Field-work.     Measure  the  three  sides, 
AB,  BC,  and  CA.     Measure  also,  as  a 

proof  line,  the  distance  from  one  of  the  cor-     ^^ p    '^ 

ners,  as  C,  to  some  point  in  the  opposite  side,  as  D,  at  which  a 
mark  should  have  been  left,  when  measuring  from  A  to  B,  at  a 
known  distance  from  A.  A  stick  or  t^dg,  with  a  slit  in  its  top,  to 
receive  a  piece  of  paper  with  the  distance  from  A  marked  on  it, 
is  the  most  convenient  mark. 

Platting.  Choose  a  suitable  scale  as  directed  in  Art.  (44). 
Then,  by  Arts.  (42)  and  (49),  draw  a  line  equal  in  length,  on  the 
chosen  scale,  to  one  of  the  sides ;  AB  for  example.  Take  in  the 
compasses  the  length  of  another  side  as  AC,  to  the  same  scale, 
and  with  one  leg  in  A  as  a  centre,  describe  an  arc  of  a  circle. 
Take  the  length  of  the  third  side  BC,  and  with  B  as  a  centre, 
describe  another  arc,  intersecting  the  first  arc  in  a  point  which  will 
be  the  tliird  corner  C.  Draw  the  lines  AC  and  BC ;  and  ABC 
will  be  the  pZrt^,  or  miniature  copy  —  as  explained  in  Art.  (35  ) — 
of  the  field  surveyed. 

Instead  of  describing  two  ares  to  get  the  point  C,  two  pairs  of 
compasses  may  be  conveniently  used.  Open  them  to  the  lengths, 
respectively,  of  the  last  two  sides.  Put  one  foot  of  each  at  the 
ends  of  the  first  side,  and  bring  their  other  feet  together,  and  their 
point  of  meeting  -vntII  mark  the  desired  third  point  of  the  triangle. 

To  "  prove  "  the  accuracy  of  the  work,  fix  the  point  D,  by  setting 
off  from  A  the  proper  distance,  and  measure  the  length  of  the  line 


CHAP.  I.] 


Suryeying  by  Diagonals. 


59 


DC,  bj  Art.  (43).     If  its  length  on  the  plat  corresponds  to  ita 
measurement  on  the  gi'ound,  the  work  is  correct.* 

Calculation.  The  content  of  the  field  may  now  be  fomid  aa 
directed  in  Art.  (65),  either  from  the  three  sides,  or  more  easily 
though  not  so  accurately,  by  measuring  on  the  plat,  by  Art.  (43), 
the  length  of  the  perpendicular  CE,  let  fall  from  any  angle  to  the 
opposite  side,  and  taking  half  the  product  of  these  two  lines. 

Examj)le  1.  Figure  49,  is  the  plat,  on  a  scale  of  two  chains 
to  one  inch,  of  a  field,  of  which  the  side  AB  is  200  links,  BC  is 
100  links,  and  AC  is  150  links.  Its  content  by  the  rule  of  Art. 
(65),  is  0.726  of  a  square  chain,  or  OA.  OR.  12P.  If  the  perpen- 
dicular CE  be  accurately  measured,  it  wiU  be  found  to  be  72^ 
links.  Half  the  product  of  this  perpendicular  by  the  base  will  be 
found  to  give  the  same  content.    • 

Ex.  2.  The  three  sides  of  a  triangular  field  are  respectively 
89.39,  54.08,  and  45.98.     Required  its  content. 

Ans.     lOOA.  OR.  lOP. 


Fig.  50. 


(91)  A  four-sided  field; 

as  Fig.  50. 

Field-ivorh.  Measure  the 
four  sides.  Measure  also 
a  diagonal,  as  AC,  thus  di- 
viding the  foui-sided  field 
into  two  triangles.  ^Nlea- 
sure  also  the  other  diagonal,  or  BD,  for  a  "  Proof  line." 

Platting.  Draw  a  line,  as  AC,  equal  in  length  to  the  diagonal, 
to  any  scale,  by  Arts.  (42)  and  (19).  On  each  side  of  it,  con- 
struct a  triangle  with  the  sides  of  the  field,  as  directed  in  the  pre- 
ceding article. 

To  prove  the  accuracy  of  the  work,  measure  on  the  plat  the 
length  of  the  "  proof  fine,"  BD,  by  Art,  (43),  and  if  it  agreea 
with  the  length  of  the  same  line  measured  on  the  ground,  the  field 
VTork  and  platting  are  both  proved  to  be  correct. 

*  It  is  a  universal  principle  in  all  surveying  operations,  that  the  work  must  be 
tested  by  some  means  independent  of  the  original  process,  and  that  the  same  re. 
suit  must  be  arrived  at  by  two  different  methods.  The  necessary  length  of  thia 
proof  line  can  also  easily  be  calculated  by  the  princii>les  of  Trigonometry- 


60 


CHAIN  SURVEYING. 


[part    1 


Calculation.  Find  the  content  of  each  triangle  st>parately,  aa 
m  the  preceding  case,  and  add  them  together ;  or,  more  brieflj, 
multiplj  either  diagonal  (the  longer  one  is  preferable)  bj  the  smii 
of  the  two  perpendiculars,  and  divide  the  product  by  two. 

Otherwise :  reduce  the  four-sided  figure  to  one  triangle  as  in 
Art.  (78)  ;  or,  use  any  of  the  methods  of  the  preceding  chapter. 

Example  3.  In  the  field  drawn  in  Fig.  50,  on  a  scale  of  3  chaina 
tc  the  inch,  AB  =  588  Hnks,  BC  =  210,  CD  =  430,  DA  =  274, 
the  diagonal  AC  =  626,  and  the  proof  diagonal  BD  =  500.  The 
total  content  will  be  lA.  OR.  17P. 

JEx.  4.  The  sides  of  a  four-sided  field  are  AB  =  12.41,  BC 
=  5.86,  CD  =  8.25,  DA  =  4.24  ;  the  diagonal  BD  =  11.55, 
and  the  proof  line  AC  =  11.04.     Required  the  content. 

Ans.  4A.  2R.  38P. 

Ux.  5.  The  sides  of  a  four-sided  field  are  as  follows  :  AB  = 
8.95,  BC  ==  5.33,  CD  =  10.10,  DA  =  6.54  ;  the  diagonal  from 
A  to  C  is  11.52  ;  the  proof  diagonal  from  B  to  D  is  10.92.  Re- 
quired the  content.  Ans. 

Mx.  6.  In  a  four-sided  field,  AB  =  7.68,  BC  =  4.09,  CD  = 
10.64,  DA  =  7.24,  AC  =  10.32,  BD  =  10.74.  Required  the 
content.  Ans. 


(92)  A  many-sided  field,  as  Fig.  51. 

Fig.  51. 
B 


cnxp.  I.J  SurYeyins  by  Diagonals.  61 

FieldrWbrk.  Measure  all  the  sides  of  the  field.  Measure 
also  diagonals  enough  to. divide  the  field  into  tnangles  ;  of  which 
there  Avill  ahvajs  be  two  less  than  the  number  of  sides.  Choose 
Buch  diagonals  as  will  divide  the  field  into  triangles  as  nearly  equi- 
lateral as  possible.  Measure  also  one  or  more  diagonals  for 
"  Proof  lines."  It  is  well  for  the  surveyor  himself  to  place  stakes 
in  advance  at  all  the  comers  of  the  field,  as  he  can  then  select  the 
best  mode  of  division. 

Platting.  Begin  with  any  diagonal  and  plat  one  triangle,  as  in 
Art.  (90).  Plat  a  second  triangle  adjoining  the  first  one,  as  in 
Ai-t.  (91).  Plat  anotker  adjacent  triangle,  and  so  proceed,  tiU  all 
have  been  laid  down  in  their  proper  places.  jNIeasure  the  proof 
lines  as  in  the  last  article. 

Calculation.  Proceed  to  calculate  the  content  of  the  figure, 
precisely  as  directed  for  the  four-sided  field,  measurmg  the  perpen- 
diculars and  calculating  the  content  of  each  triangle  in  turn ;  or 
taking  in  pairs  those  on  opposite  sides  of  the  same  diagonal ;  or 
using  some  of  the  other  methods  which  have  been  explained. 

Example  7.  Tlie  six-sided  field,  shown  in  Fig.  51,  has  the 
lengths  of  its  lines,  in  chains  and  hnks,  written  upon  them,  and  is 
divided  into  four  triangles,  by  three  diagonals.  Tlie  diagonal 
BE  is  a  "  proof-line."  The  Figure  is  drawn  to  a  scale  of  4  chains 
to  the  mch.     The  content  of  the  field  is  5A.  3R.  22P. 

Ex.  8.  In  a  five-sided  field,  the  length  of  the  sides  are  as  fol- 
lows :  AB  =  2.69,  BC  =  1.22,  CD  =  2.32,  DE  =  3.55,  EA  = 
3.23.  The  diagonals  are  AD  =  4.81,  BD  =  3.33.  Required  ita 
content.  Ans. 

(93)  A  field  may  be  divided  up  into  triangles,  not  only  by  mea^ 
Buring  diagonals  as  in  the  last  figure,  but  by  any  of  the  methoda 
shown  in  the  four  figui'es  of  Art.  (71).  The  one  which  we  have 
been  emplo3nng,  corresponds  to  the  last  of  those  figures. 

Still  another  mode  may  be  used  when  the  angles  cannot  be  seen 
from  one  another,  or  from  any  one  point  within.  Take  three  or 
more  convenient  points  within  the  field,  and  measure  from  them  tc 
the  corners,  and  thus  form  different  sets  of  triangles. 


82 


CHAIi\  SURVEYDG. 


[part  II. 


KEEPING  THE  FIELD  NOTES. 

(94)  By  Sketch.  The  most  simple  method  is  to  make  a  sketch 
of  the  field,  as  nearly  correct  as  the  unassisted  hand  and  eye  can 
produce,  and  note  down  on  it  the  lengths  of  all  the  lines,  as  in  Fig. 
61.  But  when  many  other  points  require  to  be  noted,  such  as 
where  fences,  or  roads,  or  streams  are  crossed  in  the  measurement, 
or  any  other  additional  particulars,  the  sketch  would  become  con- 
fused, and  be  hkely  to  lead  to  mistakes  in  the  subsequent  platting 
from  it.  The  follo-vving  is  therefore  the  usual  method  of  keeping 
the  Field-notes.     A  long  narrow  book  is  most  convenient  for  it. 


(95)  In  €olunms.  Draw  two  parallel  lines  about  an  inch  apart 
from  the  bottom  to  the  top  of  the  page  of  tlie 
field-book,  as  in  the  margin.  This  column,  or  pair 
of  lines,  may  be  conceived  to  represent,  the  measured 
line,  split  in  two^  its  two  halves  being  then  separated, 
an  inch  apart,  merely  for  convenience,  so  that  the 
distances  measured  along  the  line,  may  be  Avritten  be- 
tween these  halves. 

Hold  the  book  in  the  direction  of  the  measurement.  At  the 
bottom  of  the  page  write  down  the  name,  or  number,  or  letter, 
which  represents  the  station  at  which  the  survey  is  to  begin. 


A  "  station"  is  marked  with  a  triangle  or  circle,  as 
in,  the  margin.     The  latter  is  more  easily  made. 


± 


© 


In  the  complicated  cases,  which  will  be  hereafter  explained,  and 
in  which  one  long  base  line  is  measured,  and  also  many  other  sub- 
ordinate lines,  it  will  be  well,  as  a  help  to  the  memory,  to  mark 
tne  stations  on  the  Base  line  with  a  triangle,  and  the  stations  on 
the  other  lines  with  the  ordmary  circle. 

The   station   from   which   the   measure-  •      0      HoU 


ments  are  made  is  usually  put  on  the  left 
of  the  column ;  and  the  station  which  is 
measured  to,  is  put  on  the  right. 


From  A 


562 
© 


CHAP.  I.] 


Suryeying  by  Diagonals. 


68 


But  it  is  more  compact,  and  avoids  interfeiing 
with  the  notes  of  "  offsets  "  (to  be  explained  here- 
after) to  write  the  name  or  number  of  the  station 
in  the  column,  as  in  the  margin. 

Tlie  measurements  to  different  pomts  of  a  line  are 
written  above  one  another.  The  numbers  all  refer 
to  the  beginning  of  the  line,  and  are  counted  from  it. 


The  end  of  a  measured  line  is  marked  by  a  Ime 
drawn  across  the  page  above  the  numbers  wliich 
indicate  the  measurements  which  have  been  made. 

If  the  chaining  does  not  continue  along  the 
adjoining  line,  but  the  chain-men  go  to  some 
other  part  of  the  field  to  begin  another  mea- 
surement, two  lines  are  dra'SMi  across  the  page. 


B 

562 
A 

B 

400 

250 

100 

A 


"Wlien  a  line  has  been  measured,  the  marks 
r  or  ~[  are  made  to  show  whether  the  follow- 
lowing  line  turns  to  the  right  or  to  the  left. 

A  Ime  is  named,  either  by  the  names  of  the  stations  between 
which  it  is  measured,  as  the  hue  AB ;  or  by  its  length,  a  line 
562  hnks  long,  being  called  the  line  562  ;  or  it  is  recorded  as  Line 
No.  1,  Line  No.  2,  &c ;  or  as  Line  on  page  1,  2,  &c.,  of  the 
Field-book. 

"When  a  mark  is  left  at  any  pomt  of  a  line, 
ag  at  D,  in  Fig.  49,  with  the  intention  of  com- 
incr  back  to  it  ao;ain,  in  order  to  measure  to 
some  other  point,  the  place  marked  is  called  a 
False  /Station,  and  is  marked  in  the  Field-book 
F.  S. ;  or  has  a  line  drawn  around  it,  to  distin- 
guish it ;  or  has  a  station  mark  a  placed  outside 
of  the  column,  to  the  right  or  left,  according  to 
the  direction  in  which  the  measurement  from  it  is 
to  be  made..  Examples  of  these  tlu-ee  modes  are 
given  in  the  margin. 


562 

200 

0 


64 


CeAL\  SURVEYIiVG. 


[part  II. 


A  False  Station  is  named  by  its  position  on  the  line  where  it 
belongs ;  as  thus—"  200  on  562." 

When  a  gate  occurs  in  a  measured  line,  the  distance  from  the 
beginning  of  the  hne  to  the  side  of  the  gate  first  reached,  ]a  the 
one  noted. 

When  the  measured  line  crosses  a  fence,  brook, 
road,  &c.,  they  are  drawn  on  the  field-notes  in 
their  true  direction,  as  nearly  as  possible,  but 
not  in  a  continuous  line  across  the  column,  as  in 
the  first  figure  in  the  margin,  but  as  in  the  se- 
cond figure,  so  that  the  two  parts  would  form  a 


continuous  straight  line,  if  the  halves  of  the 
"  spht  line"  were  brought  together. 

It  is  convenient  to  name  the  lines,  in  the  margin,  as  being  Sides, 
Diagonals,  Proof  lines,  &c. 


(96)  The  Field-notes  of  the  triangular  field  platted  in  Fig.  49, 
are  given  below,  according  to  both  the  methods  mentioned  in  the 
preceding  Article,  pages  62  and  63. 

In  the  Field-notes  in  j^e  column  on  the  right  hand,  it  is  not  absO' 
lutely  necessary  to  repeat  the  B  and  C. 


UJ 

Z 

89 

toQ 

li. 

uFrom  D 

a. 

F.  S. 

a 

150 

to  K 

From  C 

0 

1 

yj 
a 

100 

to  C 

^From  B 

0 

1 

u 

200 

to  B 

« 

80 

F.S 

Fro7n  A 

0 

iij 

z 

c 

_i 

89 

b. 

O 

oFrom 

(^ 

on  200 

m 

A 

^    _ 
Q    1 

150 

CO 

C 

LJ 

c 

s    T 

100 
B 

B 

UJ 

200 

Q 

« 

(To^ 

CHAP.  I.] 


Sarveying  by  Diagonals. 


65 


(97)  The  Field-notes  of  the  survey  platted  in  Fig.  51,  are  given 
below.     They  begin  at  the  bottom  of  the  left  hand  column. 


Q 

F 

632 
300 
E 

Crate. 

r 

CO 

E 

662 
400 

D 

Brook. 

SIDE. 

\ 

D 

300 
270 
210 

80 
C 

Road. 

r 

SIDE. 

C 

708 
150 

B 

Grate. 

r 

u 

a 

CO 

B 

562 
A 

UJ 

1 

1 

Z 

E 

li. 

770 

o 
o 

B 

ce. 

0. 

J 

< 
z 

A 

o 

1142 

< 

1 

C 

a 

C 

-1 
< 

775 

Road, 

z 
o 
o 

/ 

480 

< 
a 

/ 

420 

1 

E 

E 

-1 
< 
z 
o 

\ 

737 
280 

\ 

< 

\ 

210 

Road. 

o 

A 

\ 

A 

270 

llj 

130 

Q 

Road. 

(0 

80 

F 

r 

CHAPTER  11.  -" 

SURVEYING  BY  TIE-LINES. 

(98)  Surveying  hy  Tie-lines  is  a  modification  of  the  method 
exDlained  in  the  last  chapter.  It  frequently  happens  that  it  is  im« 
possibb  to  measure  the  diagonals  of  a  field  of  many  sides,  in  conse- 
quence of  obstacles  to  measurements,  such  as  woods,  water,  houses, 
&c.  In  such  cases,  ^'Tie-lines,'"  (so  called  because  they  tie  the 
Bides  together),  are  employed  as  substitutes  for  diagonals. 

Thus,  in  the  four-sided  field  shown  in  the  Figure,  the  diagonals 
cannot  be  measured  because  of  woods  inter- 
vening. As  a  substitute,  measure  ofi"  from 
any  convenient  corner  of  the  field,  as  B,  any 
distances,  BE,  BF,  along  the  sides  of  the 
field.  Measure  also  the  "tie-fine"  EF. 
Measure  all  the  sides  of  the  field  as  usual. 

To  plat  this  field,  construct  the  triangle  BEF,  as  in  Art.  (90). 
Produce  the  sides  BE  and  BF,  till  they  become  respectively  equal 
to  BA  and  BC,  as  measured  on  the  ground.  Then  with  A  and  C 
as  centres,  and  with  radii  respectively  equal  to  AD  anl  CD, 
describe  arcs,  whose  intersection  will  be  D,  the  remaining  comer 
of  the  field. 

(99)  It  thus  appears  that  one  tie-line  is  sufficient  to  determine  a 
four-sided  field;  two,  a  five-sided  field,  and  so  on.  But,  as  a 
check  on  errors,  it  is  better  to  measure  a  tie-line  for  each  angle, 
and  the  agreement,  in  the  plat,  of  all  the  measurements  will  prove 
the  accuracy  of  the  whole  work. 

Smce  any  inaccuracy  in  the  length  of  a  tie-fine  is  increased  in 
proportion  to  the  greater  length  of  the  sides  which  it  fixes,  the  tie- 
Unes  should  be  measured  as  far  from  the  point  of  meeting  of  these 
sides  as  possible,  that  is,  they  should  be  as  long  as  possible. 

The  radical  defect  of  the  system  is  that  it  is  "  working  from  less 
to  greater,"  (which  is  the  exact  converse  of  the  true  principle), 
.thus  magnifying  inaccuracies  at  every  step. 


CUAP.   II.] 


Surveying^  by  Tie-lines. 


67 


A  tie-line  may  also  be  employed  as  a  "  proof  line,"  in  the  place 
of  a  diagonal,  and  tested  in  the  same  manner. 

Fig.  .53. 

If  any  angle  of  the  field  is  re-entering,  as  at 
13  in  the  figure,  measure  a  tie-line  across  the 
saUent  angle  ABC. 

(100)  Chain  Angles.  It  is  convenient,  though  not  necessary, 
to  measure  equal  distances  along  the  sides  ;  BE,  BF,  in  Fig.  52, 
and  BA,  BC,  in  Fig.  53.     "  Chain  Angles"  are  thus  formed.* 


(i---" 


(101)  Inaccessible  Areas.  The  method  of  tie-lines  can  be 
applied  to  measuring  fields  -vrhich  cannot  be  entered. 

Thus,  m  the  Figure,  ABCD  is  an  inac- 
cessible wooded  field,  of  four  sides.  To 
survey  it,  measure  all  the  sides,  and  at 
any  corner,  as  D,  measure  any  distance 
DE,  in  the  line  of  AD  produced.  Mea- 
sure also  another  distance  DF  in  the  line 
of  CD  produced.  Measure  the  tie-line  EF,  and  the  figure  can  be 
platted  as  m  the  case  of  the  field  of  Fig.  52,  the  sides  of  the  trian- 
gle being  produced  in  the  contrary  direction. 

The  same  end  would  be  attained  by  prolonguig  only  one  side,  as 
shown  at  the  angle  A  of  the  same  figure,  and  measuring  AG,  AH, 
and  GIL  It  is  better  in  both  cases  to  tie  all  the  angles  in  a 
similar  manner. 

This  method  may  be  apphed  to  a  figm'e  of  any  number  of  sides 
by  prolonging  as  many  of  them  as  are  necessary ;  all  of  them,  if 
]X)Ssible. 


*  Chain  angles  may  be  reduced  to  angles  measured  in  degrees,  by  observing 
that  the  tie-line  is  the  chord  of  the  angle  to  a  radius  equal  to  one  of  the  equal  dis- 
tances measured  on  the  sides.  Therefore,  divide  the  length  of  the  tie-line  by  the 
length  of  this  distance.  The  quotient  will  be  the  chord  of  the  angle  to  a  radius 
of  one.  In  the  Table  of  Chords,  at  the  end  of  this  volume,  find  this  quotient 
and  the  number  of  degi'ees  and  minutes  corresponding  to  it  gives  the  angle  re 
quired.  Otherwise;  since  the  chord  of  any  angle  equals  twice  the  sine  of  half 
the  angle,  we  have  this  rule  :  Divide  half  the  tie-line  by  the  measured  distance 
tiiid  in  a  table  of  natural  sines  the  angle  corresponding  to  the  quotien;,  and  mul 
tiply  this  angle  by  two,  to  get  the  angle  desired. 


CHAL\  SURVEYING. 


[part  II 


(102)  If  the  sides  CD  and  AD  were  prolonged  by  their  full 
length,  the  content  of  the  figure  could  be  calculated  without  any 
plat ;  for  the  new  triangle  DEF  would  equal  the  triangle  DAC ; 
and  the  sides  of  the  triangle  ACB  would  then  be  known. 

Fig.  55 

This  principle  may  be  extended  still  farther.  ^  t 
For  a  five-sided  field,  as  in  Fig.  55,  produce  ^n^^ 
two  pairs  of  sides,  a  distance  equal  to  their 
length,  forming  two  new  triangles,  as  shown  by 
the  dotted  fines,  and  measure  the  sides  B'D', 
and  A'D".  The  three  sides  of  each  of  these 
triangles  will  thus  be  known,  and  also  the  three 


sides  of  the  triangle  BAD, 
andBD  =  B'D'. 


since  AD  =  A'D", 


The  method  of  this  article  may  be  employed 
for  a  figure  of  six  sides  as  shown  in  Fig.  56, 
(in  which  the  dotted  lines  within  the  wooded 
field  have  their  lengths  detennined  by  the  tri- 
angles formed  outside  of  it,)  but  not  for  figures 
of  a  greater  number  of  sides. 


CHAPTER  III. 


SURVEYING  BY  PERPENDICULARS 


Bi/  the  Second  Method. 

( l«?3)  The  method  of  Surveying  hy  Perpendiculars  is  founded 
ou  the  Second  Method  of  determining  the  position  of  a  point, 
explained  in  Art.  (6).  It  is  apphed  in  two  ways,  either  to 
making  a  complete  Survey  by  "  Diagonals  and  Perpendiculars^* 
or  to  measuring  a  crooked  boundary  by  "  Off-sets.^'  Each  will  be 
considered  in  turn. 


CHAP.  Ill]  Surreying  by  Perpendiculars.  69 

The  best  metliods  of  getting  perpendiculars  on  the  ground  must, 
however,  be  fii'st  explained. 

TO  SET  OUT  PERPENDICULARS. 

(104)  Surveyor's  Cross.  The  simplest  instrument 
for  tliis  purpose  is  the  Surveyor'' s  Cross,  or  Cross-Staffs 
shown  in  the  figui-e.  It  consists  of  a  block  of  wood,  of 
any  shape,  having  in  it  two  saw-cuts,  made  very  precise- 
ly at  right  angles  to  each  other,  about  half  an  inch  deep, 
and  with  centre-bit  holes  made  at  the  bottom  of  the  cuts 
to  assist  in  findiug  the  objects.  This  block  is  fixed  on  a 
pointed  staflF,  on  which  it  can  turn  freely,  and  which 
should  be  precisely  8  links  (63|  inches)  long,  for  the 
convenience  of  short  measurements. 

To  use  the  Cross-staff  to  erect  a  perpendicular,  set  it 
at  the  point  of  the  line  at  which  a  perpendicular  is  want- 
ed. Turn  its  head  till,  on  looking  through  one  saw-cut, 
you  see  the  ends  of  the  line.  Then  will  the  other  saw- 
cut  point  out  the  direction  of  the  perpendicular,  and  thus 
guide  the  measurement  desired. 

To  find  where  a  perpendicular  to  the  line,  from  some  object,  aa 
a  corner  of  a  field,  a  tree,  &c.,  would  meet  the  fine,  set  up  the 
cross-staff  at  a  point  of  the  line  which  seems  to  the  eye  to  be  about 
the  spot.  Note  about  how  far  from  the  object  the  pei-pendicular 
at  this  pomt  strikes,  and  move  the  cross-staff  that  distance ;  and 
repeat  the  operation  tiU  the  correct  spot  is  foimd. 

(105)  To  test  the  accuracy  of  the  in-  Fig.  58.  ^  ^  ^ 
strument,  sight  through  one  sht  to  some 
point  A,  and  place  a  stake  B  in  the  line 

of  sight  of  the  other  sht.     Then  turn  its    a /tS4 

head  a  quarter  of  the  way  around,  so  (jjT 

that  the  second  sht  looked  through,  pomts  to  A.  Then  see  if  the 
other  sht  covers  B  agam,  as  it  will  if  correct.  If  it  does  not  do 
so,  but  sights  to  some  other  point,  as  B',  the  apparent  error  is 
double  the  real  one,  for  it  now  points  as  far  to  the  right  of  the  true 
point,  C  as  it  did  before  to  its  left. 


TO  CHAIN  SURVETIIVG.  U^Rx  n- 

This  is  the  first  example  we  have  had  of  the  invaluable  prin- 
ciple of  Reversion,  which  is  used  in  almost  every  test  of  the  accu- 
racy of  Surveymg  and  Astronomical  insti'uments,  its  pecuhar  merit 
being  that  it  doubles  the  real  error,  and  thus  makes  it  twice  as  easy 
to  perceive  and  correct  it. 

(106)  The  instrument,  in  its  most  finished  form,  is  made  of  a 
hollow  brass  cyluider,  which  has  two  pairs  of  shts  exactly  opposite 
to  each  other,  one  of  each  pair  being  narrow  and  the  other  wide, 
with  a  horse-hair  stretched  from  the  top  to  the  bottom  of  the  latter. 
It  is  also,  sometimes,  made  with  eight  faces,  and  two  more  pairs 
of  shts  added,  so  as  to  set  off  half  a  right  angle.  j-jg  =9 

Another  form  is  a  hollow  brass  sphere,  as  in  the 
figure.  This  enables  the  surveyor  to  set  off  perpen- 
pendiculars  on  very  steep  slopes. 


Another  form  of  the  surveyor's  cross  consists  of  two  pairs  ^  ifc'-  co 
of  plain  "  Sights,"  each  shaped  as  in  the  figure,  placed  at 
the  ends  of  two  bars  at  right  angles  to  each  other.  The 
Bht,  and  the  opening  with  a  hair  stretched  from  its  top  to 
its  bottom,  are  respectively,  at  the  top  of  one  sight  and  at 
the  bottom  of  the  opposite  sight.*  This  is  used  in  the  same 
manner  as  the  preceding  form,  but  is  less  portable  and  more  hable 
to  get  out  of  order. 

A  temporary  substitute  for  these  instruments  may  be 
made  by  sticking  four  pins  into  the  corners  of  a  square 
piece  of  board  ;  and  sighting  across  them,  in  the  direc- 
tion of  the  line  and  at  right  angles  to  it. 

(107)  Optical  Square.  The  most  convenient  and  accurate  in 
strument  is,  however,  the  Optical  Square.  The  figures  give  a  per- 
spective view  of  it,  and  also  a  plan  with  the  Hd  removed.  It  is  a 
small  circular  box,  contaimng  a  strip  of  looking-glass,  from  the 
upper  half  of  wliich  the  silverins  is  removed.     This  glass  is  placed 

*  Tilt   French  call  llie  njiiTow  opcL'ng  ailleton,  and  the  wide  one  croisee. 


CHAP.  III.] 


To  set  out  Perpendiculars 


1 


so  as  to  make  precisely  half  a  right  f"i--  ^2. 

angle  with  the  line  of  sight,  -which 
passes  through  a  sht  on  one  side 
of  the  box,  and  a  vertical  hair 
stretched  across  the  opening  on  the 
other  side,  or  a  mark  on  the  glass. 
The  box  is  held  in  the  hand  over 
the  spot  where  the  perpendicular  is 
desired,  (a  plumb  line  in  the  hand 
will  give  perfect  accuracy)  and 
the  observer  applies  his  eye  to  the 
slit  A,  looking  through  the  upper 
or  unsilvered  part  of  the  glass,  and 
turns  the  box  till  he  sees  the  other 
end  of  the  line  B,  through  the  open- 
ing C.  The  assistant,  with  a  rod, 
moves  along  in  the  direction  where  the  perpendictJar  is  desired, 
bemg  seen  in  the  silvered  parts  of  the  glass,  by  reflection  through 
the  opening  D,  till  his  rod,  at  E,  is  seen  to  coincide  with,  or  to  be 
exactly  under,  the  object  B.  Then  is  the  hne  DE  at  right  angles 
to  the  line  AB,  by  the  'optical  principle  of  the  equahty  of  the  an- 
gles of  incidence  and  reflection. 

To  find  where  a  perpendicular  from  a  distant  object  would  strike 
the  line,  walk  along  the  line,  with  the  instrument  to  the  eye,  till 
the  image  of  the  object  is  seen,  in  the  silvered  part  of  the  glass,  to 
comcide  with  the  direction  of  the  hne  seen  through  the  unsUvered 
part. 

The  instrument  may  be  tested  by  sighting  along  the  perpendicit 
lar,  and  fixing  a  point  in  the  original  line ;  on  the  principle  of 
"  Reversion." 

The  surveyor  can  make  it  for  himself,  fastening  the  glass  in  the 
box  by  four  angular  pieces  of  cork,  and  adjusting  it  by  cutting 
away  the  cork  on  one  side,  and  introducing  wedges  on  the  other 
side.     The  box  should  be  blackened  mside. 

Another  form  of  the  optical  square  contains  two  glasses,  fixed  at 
an  angle  of  45°,  and  giving  a  right  angle  on  the  principle  of  the 
Bextant. 


72  I  HAm   SURVEYING.  [part  n 

(108)  Chain  Perpendiculars, '  Perpendiculars  may  be  set  out 

with  the  chain  alone,  by  a  variety  of  methods.  These  methods 
generally  consist  in  performing  on  the  ground,  the  operations  exe* 
cuted  on  paper  in  practical  geometry,  the  chain  being  used,  in  the 
ylace  of  the  compasses,  to  describe  the  necessary  arcs. 

As  these  operations,  however,  are  less  often  used  for  the  method 
of  surveying  now  to  be  explained,  than  for  overcoming  obstacles  to 
measurement,  it  will  be  more  convenient  to  consider  them  in  that 
connection,  in  Chapter  V. 

DIAGONALS  AND  PERPENDICULARS. 

(109)  In  Chapter  I,  of  this  Part,  we  have  seen  that  plats  of  sur- 
veys made  with  the  chain  alone,  have  their  contents  most  easily 
determined  by  measuring,  on  the  plat,  the  perpendiculars  of  each 
of  the  triangles,  into  which  the  diagonals  measured  on  the  ground 
have  divided  the  field.  In  the  Method  of  Surveying  hy  Diagonals 
and  Perpendiculars,  now  to  be  explained,  the  perpendiculars  are 
measured  on  the  ground.  The  content  of  the  field  can,  therefore, 
be  found  at  once,  (by  adding  together  the  half  products  cf  each 
perpendicular  by  the  diagonal  on  which  it  is  let  fall,)  without  the 
necessity  of  previously  making  a  plat,  or  of  measuring  the  sides  of 
the  field.  Tliis  is,  therefore,  the  most  rapid  and  easy  method  of 
surveying  when  the  content  alone  is  required,  and  is  particularly 
applicable  to  the  measurement  of  the  ground  occupied  by  crops, 
for  the  purpose  of  determining  the  number  of  bushels  grown  to  the 
acre,  the  amount  to  be  paid  for  mowing  by  the  acre,  &c. 

(110)  A  three-sided  field.  Measure  the  Fig  63 
longest  side,  as  AB,  and  the  perpendicular, 
CD,  let  fall  on  it  from  the  opposite  angle  C. 
Then  the  content  is  equal  to  half  the  product 
of  the  side  by  the  perpendicular.  If  obsta-  ^  cT^  ^B 
cles  prevent  this,  find  the  point,  where  a  perpendicular  let  fall  from 
an  angle,  as  A,  to  the  opposite  side  produced,  as  BC,  would  meet 
it,  as  at  E  in  the  figure.  Then  half  the  product  of  AE  by  CB  ia 
the  content  of  the  triangle. 


CHAP    III.] 


Diagonals  and  Perpendiculars. 


73 


(111)  A  foui -sided  Oeld. 

Measure  the  diagonal  AC.  Leave 
marks  at  the  points  on  this  diago- 
nal at  which  perpendiculars  from  B  ^< 
and  from  D  would  meet  it ;  find- 
ing these  points  by  trial,  as  previ- 
ously directed  in  Arts.  (104)  and 
(107).  The  best  marks  at  these 
"False  Stations,"  have  been  described  m  Art.  (90).  Return  to 
these  false  stations  and  measure  the  perpendiculars.  "When  these 
perpendiculars  are  measured  before  finishing  the  measurement  of 
tlie  diagonal,  great  care  is  necessary  to  avoid  making  mistakes  in 
the  length  of  the  diagonal,  when  the  chammen  return  to  continue 
its  measurement.  One  check  is  to  leave  at  the  mark  as  many  pins 
as  have  been  taken  up  by  the  hind-chainman  in  coming  to  that 
point  from  the  begmning  of  the  Hne. 

Example  9.     Required  the  content  of  the  field  of  Fig.  64. 

Ans.  OA.  2R.  29P. 

The  field  may  be  platted  from  these  measurements,  if  desired, 
but  with  more  hability  to  inaccuracy  than  in  the  first  method,  in 
which  the  sides  are  measured.  The  plat  of  the  figure  is  3  chains 
to  1  inch. 

The  field-notes  may  be  taken  by  writmg  the  measurements  on  a 
sketch,  as  in  the  figure  ;  or  m  more  comphcated  cases,  by  the 
column  method,  as  below.  A  new  symbol  may  be  employed,  this 
mark,  ^-,  or  H,  to  show  the  False  Station,  from  wluch  a  perpen- 
dicular  is  to  be  measured. 

Example  10.      Calculation. 

sq.  Iks. 


g  From  200  on  480 


110 

F.S. 


toB 

H 


g  Frojn  280  on  480 


175 

F.S. 


to  J) 


From  A 


480 

280 

200 

O 


to  C 


ADC=  ^  X  480  X  175  =  42000 
sq.  chains  6.8400 
Acres  0.684 
It  is  still  easier  to  take  the  two 
triangles   together ;      multiplying 
the  diagonal  by  the  sum  of  the  per- 
pendiculars and  dividing  by  two. 


74 


€HAL\  S(JRVEYI\G. 


[part  n 


(112)  A  many-sided  field.  Fig.  65,  and  the  accompanying 
field-notes  represent  the  field  which  was  surveyed  by  the  Firsi 
Method  and  platted  in  Fig.  51. 

Fig.  65. 


From 

5.07  on  7.37 

1.54 

F.  S. 

to  i^ 

From 

1  60  on  1.1b 

2.53 
F.  S. 

to  D 

From 

5.45  o«  11.42 

4.93 
F   8. 

to  K 

From 

4.95  on  11.42 

F.  S. 

to  13 

From  E 

7.37 

5.07 

0 

to  A 

r 

H 

From  C 

7.75 

1.60 

0 

toE 

r 

From  A 

11.42 

5.45 

4.95 

0 

to  c 

Example  11.  Calculation. 
The  content  of  the  triangles  may 
be  expressed  thus : 

sq.  IJcs. 
ABC  =  1x1142x267=152457 
AEC  =  1x1142x493=281503 
CDE  =  ix  775x253=  98037 
AEF  =  ix   737x154=  56749 


sq.  chains  58.8746 

Acres  5.88746 

or,  5A.  311.  22P. 

The  first  two  triangles  nfight 

have  been  taken  together,  as  in 

the  previous  field. 

Content  calculated  from  tht 
perpendiculars  will  generally  var 
ry  slightly  from  that  obtained  bj 
measui-mg  on  the  plat. 


CHAP.  III.] 


Offsets. 


75 


(113)  A  small  field  which  has  many  sides,  may  sometimes  be 
conveniently  surveyed  by  taking  one  diagonal  and  measuring  the 
perpendiculars  let  fall  on  it  from  each  angle  of  the  field,  and  thus 
dividing  the  whole  area  into  triangles  and  trapezoids  ;  as  in  Fig.  36, 
page  48. 

The  fine  on  which  the  perpendiculars  are  to  be  let  fall,  may  also 
be  outside  of  the  field,  as  in  Fig.  37,  page  48. 

Such  a  survey  can  be  platted  very  readily,  but  the  length  of  the 
perpendiculars  renders  the  plat  less  accurate. 

This  procedure  supphes  a  transition  to  the  method  of  "  Ofisets," 
which  is  explained  in  the  next  article. 

ir 

OFFSETS.  ' 

(114)  Ofisets  are  short  perpendiculars,  measured  from  a  straight 
line,  to  the  angles  of  a  crooked  or  zigzag  line,  near  which  the  straio;ht 
Ime   runs.      Thus,  in  the  figure, 
let  ACDB  be  a  crooked  fence, 

bounding  one  side  of  a  field.    Chain   ^^^...:.\vy-.-: .v.™ r.-'y-.>sg 

along  the  straight  line  AB,  which  runs  from  one  end  of  the  fence 
to  the  other,  and,  when  opposite  each  corner,  note  the  distance 
from  the  beginning,  or  the  point  A,  and  also  measure  and  note  the 
perpendicular  distance  of  each  comer  C  and  I)  from  the  line. 
These  comers  will  then  be  "  determined"  by  the  Second  3Iet7iod, 
Art.  (6). 

The  Field-notes,  corresponding  to  Fig. 
66,  are  as  in  the  margin.  The  measure- 
ments along  the  line  are  written  in  the 
column,  as  before,  counting  from  the  be- 
ginning of  the  line,  and  the  offsets  are 
written  beside  it,  on  the  right  or  left,  oppo- 
site the  distance  at  which  they  are  taken. 
A  sketch  of  the  crooked  line  is  also  usually 
made  in  the  Field-notes,  though  not  abso- 
lutely necessary  in  so  simple  a  case  as 

this.  The  letters  C  and  D  would  not  be  used  in  practice,  but  are 
here  inserted  to  show  the  connection  between  the  Field-notes  and 
the  plat. 


0 


D|25 

C|30 

\ 

\ 

From  A  0 


300 


250 
100 


0 


toB. 


76 


CHAm  SURVEYING. 


[part  II 


In  taking  the  Field-Notes,  the  widths  of  the  offsets  should  no* 
be  drawn  proportionally  to  tlie  distances  between  them,  but  the 
breadths  should  be  greatly  exaggerated  in  proportion  to  the  lengths. 

(115)  A  more  extended  example,  with  a  little  different  notation, 
is  given  below.  In  the  figure,  which  is  on  a  scale  of  8  chains  to 
one  inch  for  the  distances  along  the  line,  the  breadths  of  the  ofiseta 
are  exaggerated  to  four  times  their  true  proportional  dimensions. 

Fis.  67. 


B 

1500 

0 

1250 

20 

0 

1000 

0 

30 

750 

50 
40 

500 
250 
0 
A 

(116)  The  plat  and  Field-notes  lA'  the  position  of  two  nouses, 
determined  by  offsets,  are  given  below  on  a  scale  of  2  chains  to  1 
inch. 

I  I  Fig.  68. 

B 


250 

to   B 

30  20 

185 
150 

Frojn  A. 

90 
50 

0 

10   ^ 

10   ^ 

30 

(117)  Double  offsets  are  sometimes  convenient;  and  sometimes 
triple  and  quadruple  ones.  Below  are  given  the  notes  and  the 
plat,  1  chain  to  1  inch,  of  a  road  of  varying  width,  both  sides  of 
which  are  determined  by  double  offsets.  It  will  be  seen  that  the 
line  AB  crosses  one  side  of  the  road  at  160  links  from  A,  and  the 
other  side  of  it  at  220. 


CHAP.  III.] 


Offsets. 


77 


Two  methods  of  keeping  the  Field-notes  are  given.  In  the  ifirst 
form,  the  offsets  to  each  side  of  the  road  are  given  separately  and 
connected  by  the  sign  +.  In  the  second  form,  the  total  distance 
of  the  second  offset  is  given,  and  the  two  measurements  connected 
by  the  word  "to."     This  is  easier  both  for  measuring  and  platting. 


Fig.  69 


B 

260 

240 

0 

220 

20 

200 

40 

180 

45 

160 

50+  0 

140 

65+  5 

120 

60+20 

100 

45+15 

80 

50+10 

60 

50+20 

40 

55+20 

20 

60+  0 

A 

30+60 
10+70 
50 
30 
10 
0 


B 

260 

240 

0 

220 

20 

200 

40 

180 

45 

160 

50  to    0 

140 

60  to   5 

120 

70  to  20 

100 

60  to  15 

80 

60  to  10 

60 

70  to  20 

40 

75  to  20 

20 

60  to    0 

A 

30  to  90 
10  to  80 
50 
30 
10 
0 


(118)  These  offsets  may  generally  be  taken  with  suflficient  accura- 
cy by  measuring  them  as  nearly  at  right  angles  to  the  base  line  as  the 
eye  can  estimate.  The  surveyor  should  stand  by  the  chain,  facing 
the  fence,  at  the  place  which  he  thinks  opposite  to  the  comer  to 
which  he  wishes  to  take  an  offset,  and  measure  "  square"  to  it  by 
the  eye,  which  a  little  practice  will  enable  him  to  do  with  much 
correctness. 


78 


CHAIN  SURVEYING. 


[part  II 


The  offsets  may  be  measured,  if  short,  with  an  Offset-staffs  a 
light  stick,  10  or  15  links  in  length,  and  divided  accordingly  ;  or 
if  they  are  long,  with  a  tape.  They  are  generally  but  a  few  links 
in  length.  A  chain's  length  should  be  the  extreme  limit,  as  laid 
down  by  the  Enghsh  "  Tithe  Commissioners,"  and  that  should  be 
employed  only  in  exceptional  cases.  When  the  "  Cross-staff"  is 
in  use,  its  divided  length  of  8  links,  renders  the  offset-staff  need- 
leas. 

When  oflfeets  are  to  be  taken,  the  method  of  chaming  to  the 
end  of  a  line,  described  in  Art.  (23),  page  21,  is  somewhat  modi- 
fied. After  the  leader  arrives  at  the  end  of  the  line,  he  should 
draw  on  the  chain  till  the  follower,  with  the  back  end  of  the  chain, 
reaches  the  last  pin  set.  This  facilitates  the  counting  of  the  links 
to  the  places  at  wliich  the  offsets  are  taken. 

The  offsets  are  to  be  taken  to  every  angle  of  the  fence  or  other 
crooked  line ;  that  is,  to  every  point  where  it  changes  its  direc- 
tion. These  angles  or  prominent  bends  can  be  best  found  by  one 
of  the  party  walking  along  the  crooked  fence  and  directing  another 
at  the  chain  what  points  to  measure  opposite  to.  If  the  line  which 
is  to  be  thus  determined  is  curved,  the  offsets  should  be  taken  to 
points  so  near  each  other,  that  the  portions  of  the  curved  line  lying 
between  them  may,  without  much  error,  be  regarded  as  straight. 
It  will  be  most  convenient,  for  the  subsequent  calculations,  to  take 
the  offsets  at  equal  distances  apart  along  the  straight  line  from 
which  they  are  measured.  . 

In  the  case  of  a  crooked  brook,  such  as  is  shown  in  the  figure 
given  below,  offsets  should  be  taken  to  the  most  prominent  angles, 
such  as  are  marked  a  a  a'm.  the  figure,  and  the  intermediate  bends 
may  be  merely  sketched  by  eye. 

Fig.  70. 


When  offsets  from  lines  measured  around  a  field  are  taken  insidf 
of  these  bounding  Imes,  they  are  sometimes  distinguished  as  Insets 


CHAP.  III.] 


Offsets. 


79 


(119)  Platting.  The  most  rapid  method  of  plattmg  the  offsets, 
is  by  the  use  of  a  Platting  Scale  (described  in  Art.  49)  and  an 
Offset  Scale,  which  is  a  short  scale  divided  on  its  edges  hke  a 
platting  scale,  but  having  its  zero  in  the  middle,  as  in  the  figure. 


Fis.  71 


The  platting  scale  is  placed  parallel  to  the  line,  with  its  zero 
point  opposite  to  the  beginning  of  the  line.  The  offset  scale  is 
sHd  along  the  platting  scale,  till  its  edge  comes  to  a  distance  on 
the  latter  at  which  an  offset  had  been  taken,  the  length  of  which  is 
marked  off  with  a  needle  point  from  the  offset  scale.  This  is  then 
shd  on  to  the  next  distance,  and  the  operation  is  repeated.  If 
one  person  reads  off  the  field-notes,  and  another  plats,  the  opera- 
tion will  be  greatly  facilitated.  The  points  thus  obtained  are 
joined  by  straight  lines,  and  a  miniature  copy  of  the  curved  line  is 
thus  obtained ;  all  the  operations  of  the  platting  being  merely  re- 
petitions of  the  measurements  made  on  the  ground. 

If  no  offset  scale  is  at  hand,  make  one  of  a  strip  of  thick  drawing 
paper,  or  pasteboard ;  or  use  the  platting  scale  itself,  turned  cross- 
ways,  having  previously  marked  off  from  it  the  points  from  which 
the  offsets  had  been  taken. 

In  plats  made  on  a  small  scale,  the  shorter  offsets  are  best  esti- 
mated by  eye. 

On  the  Ordnance  Survey  of  Ireland,  the  platting  of  offsets  is 
facilitated  by  the  use  of  a  combuiation  of  the  offset  scale  and  the 
platting  scale,  the  former  bemg  made  to  slide  in  a  groove  in  the 
latter,  at  right  angles  to  it. 


(120)  Calculating  Content.      When  the  crooked  line  deter- 
flttined  by  offsets  is  the  boundary  of  a  field,  the  content,  enclosed 


80  CHAL\  SURVEYING.  [part  ii 

Detween  it  and  the  straight  line  surveyed,  must  be  determined, 
that  it  may  be  added  to,  or  subtracted  from,  the  content  of  the 
field  bounded  by  the  straight  lines.  There  are  various  methods  of 
effecting  this. 

The  area  enclosed  between  the  straight  and  the  crooked  lines  is 
divided  up  by  the  offsets  into  triangles  and  trapezoids,  the  content 
of  which  may  be  calculated  separately  by  Arts.  (65)  and  (67), 
and  then  added  together.  The  content  of  the  plat  on  page  76, 
will,  therefore,  be  1500  +  4125  +  625  =  6250  square  links  = 
0.625  square  chain.  The  content  of  the  plat  on  page  76,  will  in 
like  manner  be  found  to  be,  on  the  left  of  the  straight  line  30,000 
square  links,  and  on  its  right  5,000  square  Hnks. 

(121)  Wlien  the  offsets  have  heen  taken  at  equal  distances,  the 
content  may  be  more  easily  obtamed  by  adding  together  half  of 
the  fii-st  and  of  the  last  offset,  and  all  the  intermediate  ones,  and 
multiplying  the  sum  by  one  of  the  equal  distances  between  the  off- 
sets.    This  rule  is  merely  an  abbreviation  of  the  preceding  one. 

Thus,  in  the  plat  of  page  76,  the  distances  bemg  equal,  the  con- 
tent of  the  offsets  on  the  left  of  the  straight  line  will  be  120  x  250 
=  30,000  square  Imks,  and  on  the  right  20  X  250  =  5,000 
square  links ;  the  same  results  as  before. 

AYhen  the  line  determined  by  the  offsets  is  a  curved  line,  "  Simp- 
son's rule"  gives  the  content  more  accurately.  To  employ  it,  an 
even  number  of  equal  distances  must  have  been  measured  in  the 
part  to  be  calculated.  Then  add  together  the  first  and  last  ofiset, 
four  times  the  sum  of  the  even  offsets,  (i.  e.  the  2d,  4th,  6th,  &c.,) 
and  twice  the  sum  of  the  odd  offsets,  (i.  e.  the  3d,  5th,  7th,  &c.,) 
not  including  the  first  and  the  last.  Multiply  the  sum  by  one  of 
the  equal  distances  between  the  offsets,  and  divide  by  3.  The 
quotient  will  be  the  area. 

Example  12.  The  offsets  from  a  straight  line  to  a  curved  fence, 
were  8,  9,  11,  15,  16,  14,  9,  links,  at  equal  distances  of  5  links. 
What  was  the  content  included  between  the  curved  fence  and  the 
Btraif^ht  Ime  ?  Ans.       371.666 


cflAP.  iii.J  Offsets.  81 

(122)  Many  erroneous  rules  have  been  given  on  this  part  of  the 
subject.  One  rule  directs  the  surveyor  to  divide  the  sum  of  all 
the  offsets  by  one  less,  than  their  number,  and  multiply  the  quotient 
by  the  whole  length  of  the  straight  line  ;  or,  what  is  the  same  thing, 
to  multijDly  the  sum  of  all  the  offsets  by  the  common  distance  be- 
tween them.  This  will  be  correct  only  when  the  offsets  at  each 
end  of  the  line  are  no  tiling,  i.  e.  when  the  curved  line  starts  from 
the  straight  hne  and  returns  to  it  at  the  beginning  and  end  of  one 
of  the  equal  distances.  In  all  other  cases  it  will  give  too  much. 
A  second  rule  directs  the  surveyor  to  divide  the  sum  of  all  the  off- 
sets by  their  nmnber,  and  then  to  multiply  the  quotient  by  the 
whole  straight  line.  This  may  give  too  much,  or  too  Uttle,  accord- 
ing to  circumstances. 

Suppose  offsets  of  10,  30,  20,  80,  50,  30,  links,  to  have  been 
taken  at  equal  distances  of  a  chain.  The  correct  content  of  the 
enclosed  space  is  200  X  100  =  2  square  chains.  The  j&rst  of  the 
above  rules  would  give  2.2  square  chains,  and  the  second  would 
give  1.8333  chams. 

(123)  Reducing  to  one  triangle  the  many-sided  figure  which  is 
formed  by  the  offsets,  is  the  method  of  calculation  sometimes  adopted. 
This  has  been  fully  explained  in  Part  I,  Art.  (78),  &c.  The 
method  of  Art.  (83)  is  best  adapted  for  this  purpose. 

(124)  Equalizing,  or  giving  and  taking ,  is  an  approximate 
mode  of  calculation  much  used  by  practical  surveyors.  A  crooked 
hne,  detennined  by  offsets,  having  been  platted,  a  straight  line  is 
di'awn  on  the  plat,  across  the  crooked  line,  leaving  as  much  space 
outside  of  the  straight  hne  as  inside  of  it,  as  nearly  as  can  be  esti- 
mated by  the  eye,  "  Equalizing"  it,  or  "  Giving  and  taking"  equal 

Fig.  72. 

I 

r 

I I 

portions.  The  straight  line  is  best  determined  by  laying  across 
the  irregular  outline  the  straight  edge  of  a  piece  of  transparent 
horn,  or  tracing  paper,  or  glass,  or  a  fine  thread  or  horee-haii 

6 


82  CHAIN  SURVEYING.  [part  il 

Btrecclied  straight  by  a  light  bow  of  whalebone.  In  practical 
hands,  this  method  is  sufficiently  accurate  in  most  cases.  The  stu- 
dent will  do  well  to  try  it  on  figures,  the  content  of  which  he  has 
previously  ascertained  by  perfectly  accurate  methods. 

Sometimes  this  method  may  be  advantageously  combined  with 
the  preceding;  short  lengths  of  the  croooked  boundary  being 
"  Equalized,"  and  the  fewer  resulting  zigzags  reduced  to  one  line 
by  the  method  of  Art.  (78),  &c. 


CHAPTER  IV. 


SURVEYING  BY  THE  PRECEDING  METHODS  COMBINED. 

125)  All  the  methods  which  have  been  explained  in  the  three 
preceding  chapters  —  Surveying  by  Diagonals,  by  Tier-lines,  and 
by  Perpendiculars,  particularly  in  the  form  of  offsets — are  fre- 
quently required  in  the  same  survey.  The  method  by  Diagonals 
should  be  the  leading  one ;  in  some  parts  of  the  survey,  obstacles 
to  the  measurement  of  diagonals  may  require  the  use  of  Tie-lines  ; 
and  if  the  fences  are  crooked,  straight  hues  are  to  be  measured 
near  them,  and  their  crooks  determined  by  Offsets. 

(126)  Offsets  are  necessary  additions  to  almost  every  other 
method  of  surveying.  In  the  smallest  field,  surveyed  by  diagonals, 
unless  all  the  fences  are  perfectly  straight  lines,  their  bends  must 
be  determined  by  offsets.  The  plat  (scale  of  1  chain  to  1  inch), 
and  field-notes,  of  such  a  case  are  given  below.     A  sufficient  num- 


CHAP.   IV.] 


Diagonals,  Tic-lines  and  Offsets. 


S3 


ber  of  the  sides,  diagonals,  and  proof-lines,  to  prove  the  work,  should 
be  platted  before  platting  the  offsets. 

Fig.  7.3. 
B  3.  60. 


0 

0 

360 

6 

315 

10 

275 

UJ 

n 

5 

215 

eS 

0 

150 

0 

115 

10 

80 

5 

65 

8 

B 

or 

B 

0 

125 

UJ 

11 

90 

Q 

00 

23 

62 

12 

22 

0 

A 

L.  . 

O  U 

O  2 

a  _i 


J 

C 

o 

If 

310 

Q 

A 

r 

A 

0 

248 

u 

11 

180 

CO 

0 

105 

0 

65 

5 

D 

0  r 

D 

0 

135 

, 

15 

110 

Q 

13 

90 

(0 

0 

50 

0 

30 

9 

C 

or 

Example  13.     Required  the  con- 
i€nt  of  the  above  field.         Ans 


(127)  Field-books.  The  difficulty  and  the  importance  of  keep 
ing  the  Field-notes  clearly  and  distinctly,  increase  with  each  new 
combination  of  methods.  For  this  reason,  three  different  methods 
of  keeping  the  Field-notes  of  the  same  survey  will  now  be  given, 
(from  Bourns'  Surveying),  and  a  careful  comparison  by  the  stu- 
dent of  the  corresponding  portions  of  each  will  be  very  profitable 
to  him. 


M 


CHAIN  SURVEYIiW. 
FIELD-BOOK  No.  1. 


[PABT  17 


FieldrBooh  No.  1  (Fig.  74)  shews  the  Sketch  method,  exfilain 
»d  in  Art.  (91). 


CHAP.  IT.]  Diagonals^  Tie-lines  and  Oflsets. 

FIELD-BOOK  IVo.  2. 


85 


\ 


(  4570     \ 

\  I 

4080 

{''^ono  *) 

3480 
3060 


("  3020  J 
1   2450l 


2300 

X 


1300 


1260 

X 

760 

.'''Sso'^ 

> ^ 

100 

620 
260 

o 
A 


ioX 


15 


/l900/ 
1300 

1250 
1200 

1020 
680 


1390 

('l230^, 
v_ ^ 

uso 

X 
700 

640 

580 

o 
vn 


lit 


200 
190 


to  A 


Field-Booh  No.  2  (Fig,  75)  shews  the  Colum'S  method,  exolai* 
ed  in  Art.  (95). 


86 


CHAIN  SURVEYING. 
FIELD-BOOK  No.  3. 


[PAM!  II 


Meld  Booh  No.  3  (Fig.  '^Q}  is  a  convenient  combination  of  ih# 
two  preceding  methods.  The  bottom  of  the  Boot  is  at  the  side  of 
tiiis  figure,  at  A. 


CHAP.  IV.] 


Diagonals,  Tie-lines  and  Oflsets. 


87 


(128)  It  will  easily  appear  from  the  sketch  of  Field-book  No.  1, 
how  much  time  and  labor  may  be  saved,  or  lost,  by  the  mamaer  ot 
doing  the  work.  Thus,  beginning  at  A,  and  measuiing  750  links, 
a  pole  should  be  left  there,  and  the  line  to  the  right  measured  tc 
17  chains,  or  C,  leaving  a  pole  at  12.30  as  a  new  starting  point  by 
and  by.  Then  from  C  measure  1 9  chains  to  A  again  ;  then  mea- 
sure from  A  to  B,  and  from  B  back  to  the  pole  left  at  7.50  on  the 
main  line. 


i- 


(129)  The  example  which  will  now  be  given  shows  part  of  the 
Field-notes,  the  plat,  (on  a  scale  of  6  inches  to  1  mile  [1 :  10,560]), 
and  a  partial  calculation  of  the  "  Filling  up"  of  a  large  triangle, 
the  angular  points  of  which  are  supposed  to  have  been  determined 
by  the  methods  of  Geodesic  Surveying.  They  should  be  well 
Btudied.* 


Fig.  77 


"  Capt.  Frome,  in  his  "  Trigonometrical  Surv'ey,"  from  which  this  exanap!* 
has  been  condensed,  remarks,  "  It  may,  perhaps,  be  thought  that  too  much  streM 
'•8  laid  on  forms  ;  but  method  is  a  most  essential  part  of  an  undertaking  of  magni* 
tuie :  and  without  excellent  preliminary  aiTangements  to  ensure  uniformity  in 
all  tLe  most  trifling  details,  the  work  never  could  go  on  creditably." 


68 


CHAIN  SURVEYnG. 


[PABT  n 


0 
34 

50 

70 

82 

From  C 


2564 


2452 


2324 


1264 


1240 

1140 

950 

772 
604 
502 
450 
342 
220 

A 

c 


80 

D 

A 

Q 

CO 
CO 

CO 
1700 

0 

1530 

84 

1420 

40 

0 

1340 

0 

0 

FrortiK 

A 

toD 

52 

A 

A 

86 

A 

100 

60 

0 

INI 

to 

3296 

0 

V      y 

3275 

54 

3120 

62 

toK 

2940 

S-o 

2572 

60 

100 

62 

42 

0 

C 

A 

o 

LO 

4050 
3890 
3730 
3540 
3420 

0 

30 

72 

2484 

S 

40 

60 

0 

2332 
2206' 
2056 
1805 
1550 

0 

40 
50 

X 

1442 

0 

FromJ) 

A 

D 

to  C. 

In  the  above  specimen  of  a  field-book,  (which  resembles  that  on 
page  85),  all  offsets,  except  those  having  relation  to  the  boundary 
lines,  are  purposely  omitted,  to  prevent  confusion,  the  example 
being  given  solely  to  illustrate  the  method  of  calculating  these 
larger  divisions.  Rough  diagrams  are  drawn  in  the  field-book  not 
to  any  scale,  but  merely  bearing  some  sort  of  resemblance  to  the 
lines  measured  on  the  ground,  for  the  purpose  of  showing,  at  any 
period  of  the  work,  their  dii'ections  and  how  they  are  to  be  connect- 
ed ;  and  also  of  eventually  assisting  in  laying  down  the  diagram 
and  content  plat.  On  these  rough  diagrams  are  written  the  li*- 
tinctive  letters  by  which  each  line  is  marked  in  the  field-book,  and  also 
its  length,  and  the  distances  between  pomts  mai'ked  upon  it,  from 
which  other  measurements  branch  off  to  connect  the  interior  por- 
tions of  the  district  surveyed. 

(130)  CalcnlationSt  The  calculation  of  one  of  the  figures,  Hfl, 
is  given  below  in  detaU.  It  is  composed  of  the  triangle  DPQ,  with 
oflfeets  along  the  sides  PQ ;  and  of  the  triangle  DWX,  with  offsets 


CUAP.  IV.] 


Diagonals,  Tie-lines  and  Offsets. 


ss 


along  the  sides  PW  and  "WX.  From  the  content  thus  obtained 
must  be  subtracted  the  offsets  on  PQ,  belonging  to  the  figure  JL, 
and  those  on  WX  belonging  to  the  figure  ji^.  When  the  offsets 
are  triangles,  (right  angled,  of  course),  the  base  and  perpendicular 
are  put  down  as  two  sides ;  when  they  are  trapezoids,  the  two 
parallel  sides  and  the  distance  between  them  occupy  the  columns 
of  "sides." 


DIVISION. 


TRIANGLE 

OR 
TRAPEZOID.  I 


1st 

SIDE. 


i^D 
SIDE. 


5D 
SIDE. 


CONTENT 

IN 
CHAINS. 


DPQ       1080  1698  1078     86.2650 


Additives. 


PQ 


DWX 
PW 

WX 


30 


1370 
30 


o2 
30 


56 
36 


1442 


56l 
36 


250 

80 

216 


770 
310 
114 
104 
90 


Total  Additives, 


.6500 
.3280 
.3240 


1.3020 
51.8339 
.4650 
.3192 
.4784 
.1620 


.9596 


140.8255 


m , 

Subtractives. 


PQ 


^\'K 


50 
30 


52 
64 


oO 
30 


52 
64 


174j 
292 


142 

232 

^88 


Total  Subtractives, 
Total  Additives, 

\         I 
Difference, 


.4350 

1.1680 

.0990 


1.7020 

.3692 

1.3456 

.2816 


1.9964 


3.6984 
140.8255 

137.1271 


90 


CHim  SURVEYIXG. 


[part  II. 


The  other  figures,  comprised  within  the  large  triangle,  are  record- 
ed and  calculated  in  a  similar  manner.     An  abridged  reoister  of 


the  results  is  given  below. 


DIVISION. 


US 


ADDITIVES. 


DNS 
and  offsets. 


SUBTRACTIVE. 


DWX 
NUV 

and  offsets. 


DIFFERENCE  IN.  ' 
SQUARE  CHAINS. 


140.4893 


DNO 

and  offsets. 


DPQ 

and  offsets. 


100.1882 


ANO 

and  offs^^'L 


NRM 

and  offsets. 


103.9778 


ID 


HTIS 

NUV 

NRM 

and  offsets. 


Offsets. 


81.6307 


CNS 
and  offsets. 


HTN 

and  offsets. 


109.5064 


M 


DPQ 

DWX 

and  offsets. 


Offsets 


137.1271 


Total, 


672.9195 


The  accuracy  of  the  preceding  calculations  of  the  separate  figures 
must  now  be  tested  by  comparing  the  sum  of  their  areas  with  that 
of  the  large  triangle  ACD,  which  comprises  them  all.  Their  area 
must  previously  be  "increased  by  the  offsets  on  the  lines  CS  and  CH, 
which  had  been  deducted  from  H,  and  which  amomit  respectively 
to  3.5270  and  2.8690.  The  total  areas  will  then  equal  679.3155 
square  chains.  That  of  the  triangle  ACD  is  679.5032  ;  a  differ- 
ence of  less  than  a  fifth  of  a  square  chain,  or  a  fiftieth  of  an  acre  ; 
or  about  one-fortieth  of  one  per  cent,  on  the  total  area. 


(131)  The  six  lines.  In  most  cases,  great  or  small,  six  fuu' 
damental  lines  will  need  to  be  measiired ;  viz.  four  approximate 
boundary  lines,  forming  a  quadrilateral,  and  its  two  diagonals. 
Small  triangles,  to  determine  prominent  points,  can  be  formed  within 
and  without  these  main  fines  by  the  First  Method,  Art.  (5), 
and  the  lesser  irregularities  can  be  determined  by  offsets. 


CHAP,  iv.l  Diagonals,  Tie-lines  and  Offsets. 

Fig.  78. 


91 


Thus,  in  the  above  figure,  two  straight  lines  AB  and  CD  are 
measured  through  the  entire  length  and  breadth  of  the  farm,  or 
township,  which  is  to  be  surveyed.  The  connecting  lines  AC,  OB, 
BD  and  DA  are  also  measured,  uniting  the  extremities  of  the 
fii'st  two  lines.  The  last  four  lines  thus  form  a  quadrilateral,  which 
is  divided  into  two  ti-iangles  by  one  of  the  first  measured  lines, 
while  the  second  serves  as  a  proof-line.  The  distance  from  the 
intersection  of  the  two  diagonals  to  the  extremities  of  each,  being 
measured  on  the  ground  and  on  the  plat,  affords  an  additional  test. 

Other  points  of  the  disti'ict  surveyed  (as  E,  G,  K.,  &c.,  in  the 
figure,)  are  determined  by  measuring  the  distances  from  them  to 
known  points  (as  M,  N,  P,  R,  &c.,  in  the  figure)  situated  on  some 
of  the  six  fundamental  lines,  thus  forming  the  triangles  T,  T. 

The  intersection  0  of  the  main  diagonals,  and  also  the  intersec- 
tions of  the  various  minor  lines  with  the  main  fines  and  with  each 
other,  should  all  be  carefully  noted,  as  additional  checks  when  the 
work  comes  to  be  platted. 


92 


CHAIN  SURVEYIXG. 


[part  II 


The  larger  figures  are  determined  first,  and  the  smaller  onea 
based  upon  them,  in  accordance  with  this  important  principle  in 
all  surveying  operations,  always  to  work  from  the  whole  to  the 
parts,  and  from  greater  to  less.  The  unavoidable  inaccuracies  are 
thus  subdivided  and  diminished.  The  opposite  course  would  accu- 
mulate and  magnify  them. 

These  additional  lines,  wliich  form  secondary  triangles,  should 
be  so  chosen  and  ranged  as  to  pass  through  and  near  as  many  ob- 
jects as  possible,  in  order  to  require  as  few  and  as  short  offsets  as 
the  position  of  the  lines  will  permit ;  the  smaller  irregularities  being 
determined  by  offsets  as  usual.  It  is  better  to  measure  too  many 
lines  than  too  few,  and  to  estabhsh  unnecessary  "  false  stations," 
rather  than  not  to  have  enough.  \ 

T 

(132)  Exceptional  cases.  The  preceding  arrangement  of  lines, 
though  in  most  cases  the  best,  may  sometimes  be  varied  with  ad- 
vantage. Unless  the  farm  surveyed  be  of  a  shape  nearly  as  broad 
as  long,  the  two  diagonals  will  cross  each  other  obliquely,  instead 
of  nearly  at  right  angles,  as  is  desirable. 


Wlien  the  fann  is  much 
longer  than  it  is  wide,  two 
systems,  of  six  lines  each, 
may  be  used  with  much 
advantage,  as  in  Fig.  79. 
Several  such  may  be  com- 
bined when  necessary. 


iys.  80. 


In  a  case  hke  that  in  Fig. 
80,  five  lines  will  be  better 
than  six,  and  -vNill  tie  one  an- 
other together,  then*  points  of 
intersection  being  carefully 
noted. 


CHAP.  IV.] 


Diasronals.  Tic-lines  and  Offsets. 


93 


Fig  81 


In  the  farm  represented  in 
Fig.  81,  the  system  of  lines 
there  shown  is  the  best,  and 
they  will  also  tie  one  another. 


(133)  Much  difficulty  will  often  be  found  m  ranging  and  mea- 
suring the  long  lines  required  by  this  method  in  extensive  surveys. 
Various  contrivances  for  overcoming  the  obstacles  which  may  be 
met  with,  will  be  explained  in  the  following  chapter.  It  will  often 
be  convenient  to  measure  the  minor  lines  along  roads,  lanes, 
paths,  &;c.,  although  they  may  not  lie  in  the  most  desirable  direc- 
tions. Steeples,  chimneys,  remarkable  trees,  and  other  objects  of 
that  character,  may  often  be  sighted  to,  and  the  hne  measured  to- 
wards them,  wnth  much  saving  of  time  and  labor.  The  pouit  where 
the  measured  lines  cross  one  another  should  always  be  noted,  and 
they  wiU  thus  form  a  very  complete  series  of  tie-lin«s.* 

A  view  of  the  district  to  be  surveyed,  taken  from  some  elevated 
position,  will  be  of  much  assistance  in  planmng  the  general  direc- 
tion of  the  lines  to  be  measured. 


(134)  Inaccessible  Areas. 

A  combination  of  offsets  and 
tie-hnes  supplies  an  easy  me- 
thod of  surveying  an  inacces- 
sible area,  such  as  a  pond, 
swamp,  forest,  block  of  houses, 
&c.,  as  appears  from  the  fi- 
gure ;  in  which  external  bound- 
ing lines  are  taken  at  will  and 


Fig.  82. 


*  To  fiud  the  exact  point  of  intersection  :/  these  lines,  which  are  only  visual 
lines,"  (explained  in  Art.  (19),)  three  persons  are  necessary:  one  stands  at  sonce 
point  of  one  of  the  lines  and  sights  to  some  other  point  on  it ;  a  second  does  the 
game  on  the  second  line  ;  by  signs  they  direct,  to  ri^ht  or  left,  the  movements  of  a 
third  person,  who  holds  a  rod,  till  he  is  placed  in  ooth  of  the  lines  and  thus  at 
tlieir  intersection,  on  the  principle  of  Art.  (11). 


94 


€HAL\   SURVEYING. 


[part  II. 


measured, and  tied  by  "tie-lines'*  measured  between  these  lines, 
prolonged  when  necessary,  as  in  Art.  (101),  while  oSsets  from 
them  deteraiine  the  irregularities  of  the  actual  boundaries  of  the 
pond,  &c. 

These  offsets  are  insets,  and  their  content  is,  of  course,  to  be 
subtracted  from  the  content  of  the  principal  figiire. 

Even  a  circular  field  might  thus  be  approximately  measured  from 
the  outside. 

If  the  shape  of  the  field  admits  of 
it,  it  will  be  preferable  to  measure 
four  lines  about  the  field  in  such 
directions  as  to  enclose  it  in  a  rect- 
angle, and  to  measure  offsets  from  the 
sides  of  this  to  the  angles  of  the 
field. 


Fig.  83. 


■#S«^J 


(135)  When  one  of  the  lines  with  which 
an  inaccessible  field  is  surrounded,  as  in 
the  last  two  figures,  cuts  a  corner  of  the 
field,  as  in  Fig.  84,  the  triangle  ABC  is 
to  be  deducted  from  the  content  of  the 
enclosing  figure,  and  the  triangle  CDE 
added  to  it.  The  triangle  DEF  is  also 
to  be  added,  and  the  triangle  FGH  de- 
ducted. To  do  this  directly,  it  would  be 
necessary  to  find  the  points  of  intersection 
C  and  F.  But  this  may  be  difficult,  and 
can  be  dispensed  with  by  obtaining  the 
difference  of  each  pair  of  triangles.  The 
difference  of  ABC  and  CDE  will  be  ob- 
tained at  once  by  multiplying  the  differ- 
ence of  the  offsets  AB  and  DE  by  half  of  BE ;  and  the  difierenca 
of  DEF  and  FGH  by  multiplymg  the  difference  of  DE  and  GH 
by  half  of  EG.* 


*  For,  making  the  triangle  Bmn  =  ABC,  then  mtiEG  =  En  X  h  {'"^^  +  ^^^ 
(DE  —  AB)  X  i  EB  ;  and  so  with  the  other  pair  of  triangles. 


CHAP,  IV.]      Diagonals,  Tie-lines  and  Perpendiculars. 


«5 


(136)  Roads.     A  winding  Road  may  also  be  surveyed  thas,  as 
18  shown  in  Fig.  85 ;  straight  lines  being  measured  in  the  road, 

Fig.  85. 


their  changes  in  direction  determined  by  tie-lines,  tying  one  line  to 
the  preceding  one  prolonged,  as  explained  in  Chapter  II,  of  this 
Part,  and  points  in  the  road-fences,  on  each  side  of  these  straight 
lines,  being  determined  by  offsets. 

A  Riyer  may  also  be  supposed  to  be  represented  by  the  above 
winding  lines ;  and  the  lower  set  of  lines,  tied  to  one  another  as 
before,  and  with  offsets  from  them  to  the  water's  edge,  will  be  suf- 
ficient for  making  an  accurate  survey  of  one  side  of  the  river. 

(137)  Towns.  A  town  could  be  surveyed  and  mapped  in  the 
same  manner,  by  measuring  straight  lines  through  all  the  streets, 
determining  their  angles  by  tie- lines,  and  taking  offsets  from  thero 
to  the  blocks  of  houses. 


9b  CHAIX  SURVEYL\G.  [part  u 


CHAPTER  V. 


OBSTACLES  TO  MEASUREMENT  L\  CHAIN  SURVEYIXG. 

(138)  In  the  practice  of  the  various  methods  of  surveying  which 
have  been  explained,  the  hills  and  valleys  which  are  to  be  crossed, 
the  sheets  of  water  which  are  to  be  passed  over,  the  woods  and 
houses  which  are  to  be  gone  through — all  these  form  obstacles  to 
the  measurement  of  the  necessary  lines  which  are  to  join  certain 
points,  or  to  be  prolonged  in  the  same  direction.  Many  special 
precautions  and  contrivances  are,  therefore,  rendered  necessary  ; 
and  the  best  methods  to  be  employed,  when  the  chain  alone  is  to 
be  used,  will  be  given  in  the  present  chapter. 

(139)  The  methods  now  to  be  given  for  overcoming  the  various 
obstacles  met  with  in  practice,  constitute  a  Lajs'd-Geometry. 
Its  problems  are  performed  on  the  ground  instead  of  on 
paper :  its  cojnjjasses  are  a  chain  fixed  at  one  end  and  free  to  swing 
around  with  the  other;  its  scale  is  the  chain  itself;  and  its  ruler 
is  the  sajne  chain  stretched  tight.  Its  advantages  are  that  its  sin- 
gle instrument,  (or  a  substitute  for  it,  such  as  a  tape,  a  rope,  &c.) 
can  be  found  anywhere  ;  and  its  only  auxiharies  are  equally  easy 
to  obtain,  bemg  a  few  straight  and  slender  rods,  and  a  plumb-Hne, 
for  which  a  pebble  suspended  by  a  thread  is  a  sufficient  substitute. 

Many  of  these  problems  require  the  employment  of  perpendicu- 
lar and  parallel  lines.  For  this  reason  we  will  commence  with  this 
class  of  Problems. 

The  Demonstrations  of  these  problems  will  be  placed  in  an  Ap- 
pendix to  this  volume,  which  will  be  the  most  convenient  arrange- 
ment for  the  two  great  classes  of  students  of  surveying ;  those  who 
wish  merely  the  practice  without  the  princii^les,  and  those  who  ynsh 
to  secure  both. 

The  elegant  "  Theory  of  Transversals"  will  be  an  important  ele- 
ment in  some  of  these  demonstrations.  All  of  them  will  constitute 
excellent  exercises  for  students. 


OHAP.  v.]  Obstacles  to  Measurement.  97 

PROBLEMS  ON  PERPENDICULARS.* 
Problem  1.     To  erect  a  perpendieular  at  any  point  of  a  Kne, 


(140)  First  3Ietliod.  Let  A  be  the 
point  at  which  a  perpendicular  to  the  line  is 
to  be  set  out.  Measure  oflF  equal  distances 
AB,  AC,  on  each  side  of  the  point.  Take 
a  portion  of  the  chain  not  quite  1^  times  as 
kng  as  AB  or  AC,  fix  one  end  of  this  at  B, 
and  describe  an  arc  with  the  other  end. 
Do  the  same  from  C.  The  intersection  of  these  arcs  will  fix  a  point 
D.  AD  will  be  the  perpendicular  required.  Repeat  the  operation 
on  the  other  side  of  the  Ime.  If  that  is  impossible,  repeat  it  on 
the  side  with  a  different  length  of  chain.  ^ 

(141)  Second  Method.    Measure  off  as  be- 
fore, equal  distances  AB,  AC,  but  each  about 

N  only  one-thu'd  of  the  chain.  Fasten  the  ends 
r-  of  the  chain  with  two  pins  at  B  and  C.  Stretch  ^" 
J.  it  out  on  one  side  of  the  line  and  put  a  pin  at  the 
middle  of  it,  D.  Do  the  same  on  the  other 
side  of  the  line,  and  set  a  pin  at  E.  Then  is  DE  a  perpendicular 
to  BC.  If  it  is  impossible  to  perform  the  operation  on  both  sides 
of  the  line,  repeat  it  on  the  same  side  with  a  different  length  of 
chain,  as  shown  by  the  Hnes  BF  and  CF  in  the  figure,  so  as  to  get 
a  second  point.  / ' '  ■'    '-- 


(142)   Other  Methods.     All  the  methods  to  be  given  for  the 
next  problem  may  be  applied  to  this. 


"  Many  of  these  methods  would  seldom  be  required  in  practice,  but  cases  some- 
times occur,  as  every  surveyor  of  much  experience  in  Field-work  has  found  to 
his  serious  inconvenience,  in  which  some  peculiarity  of  tlie  local  circumstancea 
forbids  any  of  the  usual  methods  being  applied.  In  such  cases  the  collection  here 
given  will  be  found  of  gi-eat  value. 

In  all  the  figui-es,  the  given  and  measured  lines  are  drawn  witn  fine  full  lines , 
•he  visual  lines,  or  lines  of  sight,  with  broken  lines,  and  the  lines  of  the  resuit 
with  heavy  full  lines.  The  points  which  are  centres  around  which  the  chain  ia 
swung,  are  enclosed  in  circles.  The  alphabetical  order  of  the  letters  attached  to 
the  points  shows  in  what  order  they  are  taken. 


^ 


CHAIN  SURVEYING. 


[part  II 


Problem  2.  To  erect  a  perpendicular  to  a  line  at  a  given  pointy 
when  the  point  is  at  or  near  the  end  of  the  line. 

(143)  First  Method.  Measure 
40  links  along  the  line.  Let  one  as- 
sistant hold  one  end  of  the  chain  at 
that  point ;  let  a  second  hold  the  20 
link  mark  which  is  nearest  the  other 
end,  at  the  given  point  A,  and  let  a 

third  take  the  50   Hnk  mark,   and         ^  f^-** 

tighten  the  chain,  drawing  equally  on  both  portions  of  it.  Then 
will  the^SO  Unk  mark  be  in  the  perpendicular  desired.  Repeat 
the  operation  on  the  other  side  of  the  line  so  as  to  test  the  work. 

The  above  numbers  are  the  most  easily  remembered,  but  the 
longer  the  hnes  measured  the  better  ;  and  nearly  the  whole  chain 
may  be  used,  thus :  Fix  down  the  36th  link  from  one  end  at  A, 
and  the  4th  link  from  the  same  end  on  the  line  at  B.  Fix  the 
other  end  of  the  chain  also  at  B.  Take  the  40th  link  mark  from 
this  last  end,  and  draw  the  chain  tight,  and  this  mark  wiU  be  in 
the  perpendicular  desu-ed.  The  sides  of  the  triangle  formed  by 
the  chain  will  be  24,  32  and  40. 

(144)  Otherwise  :  using  a  50  feet 
tape,  hold  the  16  feet  mark  at  A ; 
hold  the  48  feet  mark  and  the  ring- 
end  of  the  tape  together  on  the  line  ; 
take  the  28  feet  mark  of  the  tape,  and 
draw  it  tight ;  then  will  the  28  feet 
mark  be  in  the  perpendicular  desired. 


(145)  Second  Method.  Hold  one  end 
of  the  chain  at  A  and  fix  the  other  end  at  a 
point  B,  taken  at  will.  Swing  the  chain 
around  B  as  a  centre,  tiU  it  again  meets  the 
line  at  C.  Then  carry  the  same  end  around 
(the  other  end  remaining  at  B)  till  it  comes 
m  the  line  of  CB  at  D.  AD  is  the  perpen- 
dicular required.   C  A  J)  =  <»^^  u  .    ,.  ,:[.,,- 

V 


Fiff.  90, 


CHAP.  V.J 


Oostacles  to  Measurement, 


/u.:v:   Ca 


CB.'. 


(146)  ^A^Vc^  3Iethod.     Let  A  be  the  given 

\  point.     Choose  anj  point  B.     Measure  BA. 

'  Set  off,  on  the  given  line,  AC  =  AB.     On  CB 

2  AC2 
produced  set  off  from  C,  a  distance  =  . 

This  will  fix  the  point  D,  and  AD  will  be  the 
perpendicular  required. 

(147)  Fourth  3Iethod.  From  the 
given  pouat  A  set  off  on  the  given  line 
any  distance  AB.  From  B,  in  any 
convenient  direction,  set  off  BC  =  AB. 
Then  on  the  given  line,  set  off  AD  = 

AC.  On  CB  prolonged,  set  off  CE  = 

AD.  Join  DE  ;  and  on  DE,  from  D,  set  off  DF  =  2  AB.     Then'^^^^^fe 
will  the  line  AF  be  perpendicular  to  the  line  AD  at  the  point  AA  l\QrJcoJli^  u  •' 

Problem  3.  To  erect  a  perpendicular  to  an  inaccessible  line^ 
at  a  given  point  of  it. 

(148)  First  Metliod.  Get  points  in  the  direction  of  the  inac- 
cessible line  prolonged,  and  from  them  set  out  a  parallel  to  the  line, 
by  methods  which  are  given  in  Art.  (165),  &c.  Find  by  trial  the 
point  in  which  a  perpendicular  to  this  second  line  (and  therefore  to 
the  first  line)  will  pass  through  the  required  point. 

(149)  Second  Metliod.  If  the  line  is  not  only  inaccessible, 
but  cannot  have  its  direction  prolonged,  the  desired  perpendicular 
can  be  obtained  only  by  a  complicated  trigonometrical  operation. 


.^ 


'i(in%9 


Problem  4. 

a  given  line. 


To  let  fall  a  perpendicular  from  a  given  point  t« 


(150)  First  Method.  Let  P  be 
the  given  point,  and  AB  the  given 
line.  Measure  some  distance,  a  chain 
or  less,  from  C  to  P,  and  then  fix  one 
end  of  the  chain  at  P,  and  swing  it 
around  till  the  same  distance  meets 


100 


CHAIN  SURVEYING. 


[PAKT  II. 


the  line  at  some  point  D.  The  middle  point  E  of  the  distance  CD 
will  be  the  required  point,  at  which  the  perpendicular  from  P 
would  meet  the  line. 


(151)  Second  Method.  Stretch  a  chain,  or  a  portion  of  it, 
from  the  given  point  P,  to  some  point,  as  A,  of  the 
given  line.  Hold  the  end  of  the  distance  at  A, 
and  swing  round  the  other  end  of  the  chain 
from  P,  so  as  to  set  off  the  same  distance  along 
the  given  hne  from  A  to  some  point  B.  Mea- 
sure BP 

BP2 

desired  perpendicular  =  o~rR 


C       B 

Then  will  the  distance  BC  from  B  to  the  foot  of  the 


(152)  OtJier  Methods.     All   the  methods  given  m  the  next 
problem  can  be  applied  to  this  one. 

Problem  5.     To  let  fall  a  perpendicular  to  a  line  ^  from  a  point 
nearly  opposite  to  the  end  of  the  line. 

(153)  First  Method.  Stretch  a  chain  from  the  given  point  P, 
to  some  point,  as  A,  of  the  ^ven  line.  Fix  to 
the  ground  the  middle  pomt  B  of  the  chain 
AP,  and  swing  around  the  end  which  was  at 
P,  or  at  A,  till  it  meets  the  given  line  in  a 
point  C,  which  will  be  the  foot  of  the  re- 
quired perpendicular. 

(154)  Second  Method.  Take  any  point, 
as  A,  on  the  given  line.  Measure  a  dis- 
tance AB.  Let  the  end  of  this  distance 
on  the  chain  be  held  at  B,  and  swing  around 
the  end  of  the  chain,  till  it  comes  in  the  -^  b 
line  of  AP  at  some  point  C,  thus  making  BC  =  AB.  Measure 
AC  and  AP.     Then  the  distance  AD,  from  A  to  the  foot  of  the 

,.    ,  .     ,       APxAC 

perpendicular  reqmred  =  ■  ^  . -p    . 


CH4>P.  v.] 


Obstacles  to  Mcasnrement. 


101 


(155)  Third  Method.  At  any  convenient 
point,  as  A,  of  the  given  line,  erect  a  perpen- 
dicular, of  any  convenient  length,  as  AB,  and 
mark  a  point  C  on  the  given  line,  in  the  line 
of  P  and  B.  Measure  CA,  CB  and  CP. 
Then  the  distance  from  C  to  the  foot  of  the 

perpendicular,  i.  e.  CD  =  — ^^ — . 


Problem  6.      To  let  fall  a  perpendicular  to  a  line,  from  an 
inaccessible  point. 


(156)  First  Method.  Let  P  be  the  given 
point.  At  any  point  A,  on  the  given  line,  set  out 
a  perpendicular  AB  of  any  convenient  length. 
Prolong  it  on  the  other  side  of  the  Une  the 
same  distance.  Mark  on  the  given  line  a 
point  D  in  the  line  of  PB ;  and  a  point  E  in 
the  hne  of  PC.  Mark  the  point  F  at  the  in- 
tersection of  DC  and  BE  prolonged.  The  line 
FP  is  the  hne  required,  being  perpendicular 
to  the  given  Ime  at  the  point  G. 


Fig.  99 


(157)  Second  Method.  Let  A  and  B 
be  two  points  of  the  given  line.  From  A 
let  fall  a  perpendicular,  AC,  to  the  visual 
line  BP ;  and  from  B  let  fall  a  perpendi- 
cular, BD,  to  the  visual  line  AP.  Find 
the  point  at  which  these  perpendiculars 
intersect,  as  at  E  (see  Art.(133)),  and  the 
line  PE,  prolonged  to  F,  will  give  the 
perpendicular  required. 


Problem  7.     To  let  fall  a  perpendicular  from  a  given  point  to 
on  inaccessible  line. 

JOHI^  S.  PRELL 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 


102  CHAIN  SURVEYING.  Fpaut  ri 

(158)  First  Method.  Let  P  be 
the  given  point  and  AB  the  given 
line.  Bj  the  preceding  problem,  let 
fall  perpendiculars  from  A  to  BP,  at 
C ;  and  from  B  to  AP,  at  D ;  the 
line  PE,  passing  from  the  given  point 
to  the  intersection  of  these  perpendiculars,  is  the  desired 'perpendi* 
cular  to  the  inaccessible  line  AB. 

This  method  will  apply  when  only  two  points  of  the  line  are 
visible. 

(159)  Secorul  Method.  Through  the  given  point,  set  out,  by 
the  methods  of  Art.  (165),  &c.,  a  line  parallel  to  the  inaccessible 
line.  At  the  given  point  erect  a  perpendicular  to  the  parallel  line, 
and  it  will  be  the  required  perpendicular  to  the  inaccessible  line. 

PROBLEMS  ON  PARALLELS. 

ProlJleni  1 ,  To  run  a  line,  from  a  given  point,  parallel  to  a 
given  line. 

(160)  Mrst  Method.  Let  fall  a  perpendicular  from  the  point 
to  the  line.  At  another  point  of  the  line,  as  far  off  as  possible, 
erect  a  perpendicular,  equal  in  length  to  the  one  just  let  fall.  The 
line  joining  the  end  of  this  line  to  the  given  point  will  be  the  paral- 
lel required. 

(Ul)  Second  Method.     LetABbe  Fig.  loi. 

the  given  line,  and  P  the  given  point,  a  -- 
Take  any  point,  as  C,  on  the  given  line, 
and  from  it  set  off  equal  distances,  as 
long  as  possible,  CD  on  the  given  line, 
and  CE,  on  the  line  CP.  Measure 
DE.  From  P  set  off  PF  =  CE  ;  and  from  F,  with  a  distance  = 
DE,  and  from  P,  with  a  distance  =  CD,  describe  arcs  intersect- 
mg  in  G.  PG  will  be  the  parallel  required.  K  it  is  more  con- 
venient, PC  may  be  prolonged,  and  the  equal  triangle,  CDE,  be 
formed  on  the  other  side  of  the  line  AB. 


1 


CHAT,  v.]  Obstacles  to  Measurement.  103 

(162)  Third  3IetJwd.  Measure  from  t^'g- 102. 
P  to  any  point,  as  C,  of  the  given  line,  and  A" 
put  a  mark  at  the  middle  point,  D,  of  that 
line.  From  any  point,  as  E,  of  the  given 
line,  measure  a  line  to  the  point  D,  and  con- 
tinue it  till  DF  =  DE,  Then  will  the  line 
PF  be  parallel  to  AB. 

(163)  Fourth  Method.  Measure  from 
P  to  ajay  point  C,  of  the  given  line,  and 
continue  the  measurement  till  CD  =  CP. 
From  D  measure  to  any  pouit  E  of  the 
given  Hne,  and  continue  the  measurement 
tUl  EF  =  ED.  Then  will  the  line  PF 
be  parallel  to  AB.  If  more  convenient, 
CD  may  be  made  one-half,  or  any  other  fraction,  of  CP,  and  EF 
be  then  made  twice,  &c.,  DE. 


Fig.  104. 

c 


E 


X    \  Id--'    / 


(164)  Fifth  Method.  From  any 
rl.point,  as  C,  of  the  line,  set  off  equal  A- 
distances  along  the  line,  to  D  and  E. 
Take  a  pomt  F,  m  the  line  of  PD. 
Stake  out  the  lines  FC  and  FE,  and 
also  the  line  EP,  crossing  the  Hne  CF 

m  the  point  G.     Lastly,  prolong  the  line  DG,  till  it  meets  the 
Ime  EF  in  the  point  H.     PH  is  the  parallel  required. 

Problem  2.     To  run  a  line  from  a  given  point  parallel  to  an 

inaccessible  line.  -  -      '~ 


F 


(165)  First  3Iethod.  Let  AB 
be  the  given  line,  and  P  the  given 
point.  Set  a  stake  at  C,  in  the  line 
of  PA,  and  another  at  any  conven- 
ient point,  D.  Through  P,  set  out, 
by  the  preceding  problem,  a  parallel 
to  DA,  and  set  a  stake  at  the  point, 


tis.  105. 


^  ^  CD.",  cri  w/t 
B 


\>5: 


\ 


F^ 


-^P 


as  E,  where  this  parallel  intersects  DC  prolonged.     Through  B 


104  CHAIN  SURVEYING.  [part  ii 

Bet  out  a  parallel  to  BD,  and  set  a  stake  at  the  point  F,  where  thia 
parallel  intersects  BC  prolonged      PF  is  the  parallel  required 

(166)  Second  Method.  Set  a  stake 
at  any  point,  C,  m  the  line  of  AP,  and 
another  at  any  convenient  place,  as  at  D. 

'  Through  P  set  out  a  parallel  to  AD, 
intersecting  CD  in  E.  Through  E  set 
out  a  parallel  to  DB,  intersecting  CB  in 
F.  The  line  PF  will  be  the  parallel  re- 
quired. 

(167)  Alinement  and  Measnrement.  We  are  now  prepared, 
having  secured  a  variety  of  methods  for  setting  out  Perpendi:ular3 
and  Parallels  in  every  probable  case,  to  take  up  the  general  sub- 
ject of  overcoming  Obstacles  to  Measurement. 

Before  a  line  can  be  measured,  its  direction  must  be  determined. 
This  operation  is  called  Ranging  the  line ;  or  Alining  it ;  or 
Boning  it.*  The  word  Alinement\  will  be  found  very  convenient 
for  expressing  the  direction  of  a  line  on  the  ground,  whether 
between  two  points,  or  in  their  direction  prolonged. 

This  branch  of  our  subject  naturally  divides  itself  into  two  parts, 
the  first  of  which  is  preliminary  to  the  second  ;  viz  : 

I.  Of  Obstacles  to  Alinement ;  or  how  to  establish  the  direc- 
tion of  a  line  in  any  situation. 

II.  Of  Obstacles  to  Measurement ;  or  how  to  find  the  length  of 
a  line  which  cannot  he  actually  measured. 

I.    OBSTACLES  TO  ALINEMENT. 

(168)  All  the  cases  which  can  occur  under  this  head,  may  be 
reduced  to  two ;  viz : 

A.  To  find  pomts  in  a  line  beyond  the  given  points,  i.  e.  to 
prolong  the  line. 

B.  To  find  points  in  a  fine  between  two  given  points  of  it,  i.  e. 
U)  interpolate  points  in  the  line. 

*  This  word,  like  many  others  used  in  En^neering,  is  derived  from  a  French 
word,  Bomer,  to  mark  out,  or  limit ;  indicating  that  the  Normans  iuti'oduced  th* 
trt  of  Surveying  into  England. 

t  Slightly  modified  from  the  French  Alisr*ement. 


ooAP.  v.l  Obstacles  to  Measurement,  105 

A.    TO  PROLONG  A  LINE 

(169)  By  ranging  with  rods.  When  two  pomts  ia  i  line  are 
given,  and  it  is  desired  to  J'^'g-  ^07. 

prolong  the  line  bj  ranging    . f-iS^r^S^S^ill.llfa^i^^^!!^. 

it  out  with  rods,  three  per-  ^^  '^'S^^^SuXl^^^ 
sons  are  required,  each  furnished  with  a  straight  slender  rod,  and 
with  a  plumb-line,  or  other  means  of  keeping  their  rods  vertical. 
One  holds  his  rod  at  one  of  the  given  points,  A,  in  the  figure,  and 
smother  at  B.  A  third,  C,  goes  forward  as  far  as  he  can  Avithout 
losing  sight  of  the  fii'st  two  rods,  and  thoL,  looking  back,  puts  him- 
self "  in  line"  with  A  and  B,  i.  e.  so  that  when  his  eve  is  placed 
at  C,  the  rod  at  B  hides  or  covers  the  rod  at  A.  This  he  can  do 
most  accui-atelj  by  holding  a  plumb-line  before  his  eye,  so  that  it 
shall  cover  the  fii-^st  two  rods.  The  lower  end  of  the  plumb-bob 
will  then  indicate  the  point  where  the  thh-d  rod  should  be  placed ; 
and  io  with  the  rest.  The  first  man,  at  A,  is  then  signalled,  and 
comes  forward,  passes  both  the  others,  and  puts  himself  at  D,  "  in 
fine"  with  C  and  B.  The  man  at  B,  then  goes  on  to  E,  and  "  lines" 
himself  with  D  and  C  :  and  so  they  proceed,  in  this  "  hand  over 
hand"  operation,  as  far  as  is  desired.  Stakes  are  driven  at  each 
pouit  in  the  fine,  as  soon  as  it  is  determined. 

(170)  The  rods  should  be  perfectly  straight,  either  cylindrical  or 
polygonal,  and  as  slender  as  they  can  be  without  bending.  They 
should  be  painted  in  alternate  bands  of  red  and  white,  each  a  foot, 
or  hnk,  in  length.  Their  lower  ends  should  be  pointed  with  u'on, 
and  a  projecting  bolt  of  iron  wiU  enable  them  to  be  pressed  down 
by  the  foot  into  the  earth,  so  that  they  can  stand  alone.  When 
this  is  done,  one  man  can  range  out  a  line.  A  rod  can  be  set  per- 
fectly vertical,  by  holding  a  plumb-line  before  the  eye  at  some  dis- 
tance from  the  rod,  and  adjusting  the  rod  so  that  the  plumb-line 
covers  it  from  top  to  bottom  ;  and  then  repeatmg  the  operation  in 
a  direction  at  right  angles  to  the  former.  A  stone  dropped  froa 
top  to  bottom  of  the  rods  will  approximately  attain  the  same  end. 

When  the  lines  to  be  ranged  are  long,  and  great  accuracy  is  re 
quired,  the  rods  may  have  attached  to  them  plates  of  tin  with  oper 


106 


CHAIN  SURVEYmC. 


[part  u. 


mc's  cut  out  of  them,  and  black  horse-hairs  stretched  from  Fig.  108 
top  to  bottom  of  the  openmgs.     A  small  telescope  must 
then  be  used  for  ranging  these  hairs  m  Ime.     In  a  hasty 
emrvej,  straight  twigs,  with  their  tops  spht  to  receive  a  par 
per  folded  as  in  the  figure,  may  be  used. 

(171)  By  perpendiculars.  Fi-.  loo 

The  straight  Une,  AB  in  the  ----^ '^[  @p f — 

figure,  is  supposed  to  be  stop-       c  D        E  P 

ped  by  a  tree,  a  house,  or  other  obstacle,  and  it  is  desu'ed  to  pro- 
long tho  line  beyond  this  obstacle.  From  any  two  points,  as  A 
and  B,  of  the  line,  set  off  (by  some  of  the  methods  which  have  been 
given)  equal  perpendiculars,  AC  and  BD,long  enough  to  pass  the 
obstacle.  Prolong  this  line  beyond  the  obstacle,  and  from  any  two 
pomts  in  it,  as  E  and  F,  measure  the  perpendiculars  EG  and 
FH,  eaual  to  the  first  two,  but  in  a  contrary  direction.  Then 
will  G  and  H  be  two  points  in  the  line  AB  prolonged,  which  can 
be  continued  by  the  method  of  the  last  article.  The  points  A 
and  B  should  be  taken  as  far  apart  as  possible,  as  should  also 
the  points  E  and  F.  Three  or  more  perpendiculars,  on  each 
side  of  the  obstacle,  may  be  set  off,  in  order  to  increase  the  accu- 
racy of  the  operation.  The  same  thing  may  also  be  done  on  the 
other  side  of  the  hne,  as  another  confirmation,  or  test,  of  the  accu- 
racy of  the  prolonged  hne. 


Vv'- 


(172)  By  equilateral  triangles. 

The  obstacles,  noticed  in  the  .last  arti- 
cle, may  also  be  overcome  by  means  of 
three  equilateral  triangles,  formed  by 
the  chain.  Fix  one  end  of  the  chain, 
and  also  the  end  of  the  first  Hnk  from 
its  other  end,  at  B  ;  fijs  the  end  of  the 
33d  hnk  at  A ;  take  hold  of  the  66th 
link,  and  draw  the  chain  tight,  pulling  equally  on  each  part,  and 
put  a  pin  at  the  point  thus  fomid,  C,  in  the  figure.  An  equilateral 
triangle  will  thus  be  formed,  each  side  being  33  Imks.  Prolong 
the  line  AC,  past  the  obstacle,  to  some  point,  as  T).     Make  another 


CHAP.   \  ] 


Obstacles  to  Measurement. 


107 


equilateral  triangle,  DEF,  as  before,  and  thus  j5x  the  point  F.  Pro- 
long DF,  to  a  length  equal  to  that  of  AD,  and  thus  fix  a  point  G. 
At  G  form  a  third  equilateral  triangle  GHK,  and  thus  fix  a  point 
K.     Then  wiU  KG  give  the  direction  of  AB  prolonged. 


Let  AB  be  the  line  to  !>« 

Fig.  111. 


(173)  By  symmetrical  triangles 

prolonged.  Take  any  conve- 
nient point,  as  C.  Eangc  A 
out  the  Ime  AC,  to  a  point 
A",  such  that  CA'  =  CA. 
Range  out  CB,  so  that  CB' 
=  CB.  Range  backwards 
A'B',  to  some  point  D,  such 
that  DC  prolonged  will  pass 
the  obstacle.  Find,  by  ranging,  the  intersection,  at  E,  of  DB  and 
AC.  From  C,  measure,  on  CA',  the  distance  CE'=  CE.  Then 
range  out  DC  and  B'E'  to  their  intersection  in  P,  which  will  be  a 
required  point  in  the  direction  of  AB  prolonged.  The  symmetri- 
cal points  are  marked  by  corresponding  letters.  Several  other 
points  should  be  obtained  in  the  same  manner. 

In  this,  as  in  aU  similar  operations,  very  acute   intersections 
should  be  avoided  as  far  as  possible. 


112. 


(174)  By  transrersals.  LetABbe 
the  given  line.  Take  any  two  pomts  C 
and  D,  such  that  the  line  CD  wUl  pass 
the  obstacle.  Take  another  point,  E, 
in  the  intersection  of  CA  and  DB. 
Measure  AE,  AC,  CD,  BD  and  BE. 
Then  the  distance  from  D  to  P,  a  point 
in  the  required  prolongation,  ^^-ill  be 
DP  —  CDxEDxAE 
~  BExAC— BDxAE' 

Other  points  in  the  prolongation  may 
be   obtained  in  the   same   manner,  by 
merely  moving  the  single  point  C,  in  the 
line  of  EA ;    in  which  case  the  new  distances  CA  and  CD  wiD 
alone  require  to  be  measured. 


108  CHAIN  SCRVEYIXG.  Ipari  ii. 

^      CDxBD 
If  AE  be  made  equal  to  AC,  then  is  DP  =  j^^_^]j' 

^^      CDxAE 
If  BE  be  made  equal  to  BD,  then  is  DP  =-  ^^jZZXe" 

x^he  minus  sign  in  the'  denommators  must  be  understood  aa  only 
moamiig  that  the  diflference  of  the  two  tenns  is  to  be  taken,  without 
regard  to  which  is  the  greater. 

-^ 

(175)  By  harmonic  conjugates.  Fi^:.  us. 

Let  AB  be  the  given  Ime.     Set  a    .  nT^^'^^'^-  t* 

stake  at  any  pouit  C.    Set  stakes  at     --~.,^^  /r^-^p^y:y 

pomts,  D,  on  the  line  CA,  and  at      '  \^    '^--.^^J^''  ;  c;^-''  / 
Ej  on  the  line  CB ;  these  pomts,  ^\^       /  'y^E      /" 

D  and  E,  being  so  chosen  that  the  h<^^'*  '  1     '''' 

line  DE  will  pass  beyond  the  obsta-  'n,^  '"^fii^lii 

cle.     Set  a  fourth  stake,  F,  at  the  \iM 

intersection  of  the  lines  AE  and  "\\i 

DB.     Set  a  fifth  stake,  G,  any-  c 

where  in  the  line  CF ;  a  sixth  stake,  H,  at  the  intersection  of  CB 
and  DG  prolonged ;  and  a  seventh,  K,  at  the  intersection  of  CA 
and  EG  prolonged.  Finally,  range  out  the  lines  DE  and  KH, 
and  their  intersoction  at  P,  will  be  in  the  line  AB  prolonged. 

(176)  liy  the  complete  quadriiateral.  Let  AB  be  the  given 
hue.     Take  any  conven  Fig.  lu 

lent   point    C ;     measure  F      C_^ _^G 

from  it  to  B,  and  onward,        \>-..^ 
"m  the  same  line  prolong-  \.^^ 

ed,  an  equal  distance  to  D. 
Take  any  other  convenient 
point,  E,  such  that  CE  and 

DE  produced  wiU  clear  the  obstacle.  Measure  from  E  to  A,  and 
onward,  an  equal  distance,  to  F.  Range  out  the  lines  FC  and  DE 
to  their  intersection  in  G.  Range  out  FD  and  CE  to  inter- 
sect in  H.  Measure  GH.  Its  middle  pomt,  P,  is  the  required 
point  in  the  line  of  AB  prolonged.  The  unavoidable  acute  inter- 
Bections  in  this  construction  are  objectionable.  ,i^' 


y/ 


CMAP    V.J 


Obstacles  to  Measurement. 


109 


B.  TO  LWERPOLATE  POINTS  IN  A  LINE. 

(177)  The  most  distant  given  point  of  the  line  must  be  made 
as  conspicuous  as  possible,  by  any  efficient  means,  such  as  placing 
there  a  staflf,  beai-ing  a  flag;  red  and  wliite,  if  seen  againat 
woods,  or  other  dark  back-ground ;  and  red  and  green,  if  seon 
against  the  sky. 

A  convenient  and  portable  signal  is  sliown  m  the  figure. 

Fig.  115. 
front  View  Side  View.  Back  View. 


< 


V 

The  figure  represents  a  disc  of  tin,  about  six  inches  in  diameter, 
painted  white  and  hinged  in  the  middle,  to  make  it  more  portable. 
It  is  kept  open  by  the  bar,  B,  being  turned  into  the  catch,  C. 
A  screw,  S,  holds  the  disc  in  a  slit  in  the  top  of  the  pole. 

Another  contrivance  is  a  strip  of  tin,  which  has  its  ends  bent 
horizontally  in  contrary  directions.  As  the  wind  wiU  take  strong- 
est hold  of  the  side  which  is  concave  towards  it,  the  bent  strip  will 
continually  revolve,  and  thus  be  very  conspicuous.  Its  upper  half 
should  be  painted  red  and  its  lower  half  white. 

A  bright  tin  gone  set  on  the  staff,  can  be  seen  at  a  great  distance 
when  the  sun  is  shining. 

178)  Ranging  to  a  point,  thus  made  conspicuous,  is  very  aim- 
pie  when  the  ground  is  level.  The  surveyor  places  his  eye  at  the 
nearest  end  of  the  line,  or  stands  a  Uttle  behind  a  rod  placed  on  it, 
and  by  signs  moves  an  assistant,  holding  a  rod  at  some  point  aa 
nearly  in  the  desired  line  as  he  can  guess,  to  the  right  or  left,  till 
his  rod  appears  to  cover  the  distant  noint 


110 


CHAIIV  SURVEYING. 


[part  II 


(179)  Across  a  Talley.     When  a  valley,  or  low  spot,  intop 
venes  between  the  two  ends  Fig.  lie. 

of  the  line,  A  and  Z  m  the  |^-r 
figure,  a  rod  held  in  the 
low  place,  as  at  B,  would 
seldom  be  high  enough  to 
be  seen,  from  A,  to  cover 
the  distant  rod  at  Z.  In 
such  a  case,  the  surveyor  at  A  should  held  up  a  plumb-hne  over 
the  point,  at  arm's  length,  and  place  nis  eye  so  that  the  plumb-line 
covers  the  rod  at  Z.  He  should  then  direct  the  rod  held  at  B  to 
be  moved  till  it  too  is  covered  by  the  plumb-line.  The  point  B  is 
then  said  to  be  "  in  hne"  between  A  and  Z.  In  geometrical  lan- 
guage, B  has  now  been  placed  in  the  vertical  plane  determined  by 
the  vertical  plumb-line  and  the  pomt  Z.  Any  number  of  interme- 
diate points  can  thus  be  "  interpolated,"  or  placed  in  line  between 
A  and  Z. 

(180)  Over  a  hill.     Wlien  a  liill  rises  between  two  pouits  and 
prevents  one  being  seen  from  the  other,  as  m  the  figure,  (the  upper 

Fig.  117. 


of  which  shows  the  hill  in  "  Elevation,"  and  the  lower  part  in 
"  Plan"),  two  observers,  B  and  C,  each  holding  a  rod,  may  place 
themselves  on  the  ridge,  in  the  line  between  the  two  points,  aa 
nearly  as  tliey  can  guess,  and  so  that  each  can  at  once  see  the  other 
and  the  pouit  beyond  him.     B  looks  to  Z,  and  by  signals  puts  0 


CHAP,  v.] 


Obstacles  to  I^easiirement. 


Ill 


"in  line."  C  then  looks  to  A,  and  puts  B  in  line  at  B'.  B  re- 
peats his  operation  from  B',  putting  C  at  C,  and  is  then  himself 
moved  to  B',  and  so  they  alternately  "line"  each  other,  continu- 
ally approximating  to  the  straight  line  between  A  and  Z,  till  they 

at  last  find  themselves  both  exactly  in  it,  at  W"  and  C". 

(181)  A  single  person  may  put  himself  in  hne  between  two 
points,  on  the  same  principle,  by  laying  a  straight  stick  on  some 
support,  going  to  each  end  of  it  in  turn,  and  making  it  point  suc- 
cessively to  each  end  of  the  hne.  Thj  "  Surveyor's  Cross,"  Art. 
(104),  is  convenient  for  this  purpose,  when  set  up  between  the  two 
given  points,  and  moved  again  and  again,  until,  by  repeated  trials, 
one  of  its  shts  sights  to  the  given  points  when  looked  through  in 
either  direction. 


(182)  On  water,  A  simple  mstru- 
ment  for  the  same  object,  is  represented 
in  the  figure.  AB  and  CD  are  two 
tubes,  about  1|  inches  in  diameter,  con- 
nected by  a  smaller  tube  EF.  A  piece 
of  looking-glass,  GH,  is  placed  in  the 
lower  part  of  the  tube  AB,  and  another, 
KL,  in  the  tube  CD.  The  planes  of  <^ij 
the  two  mirrors  are  at  right  angles  to 
each  other.  The  eye  is  placed  at  A,  and 
the  tube  AB  is  directed  to  any  distant 
object,  as  X,  and  any  other  object  be- 
hind the  observer,  as  Z,  will  be  seen,  ap- 
parently under  the  first  object  in  the  mirror  GH,  by  reflection  from 
the  mirror  KL,  when  the  observer  has  succeeded  in  getting  in  line 
between  the  two  objects.  M,  N,  are  screws  by  which  the  mirror 
KL  may  be  adjusted.  The  distance  between  the  two  tubes  will 
cause  a  small  parallax,  which  will,  however,  be  insensible  except 
whec  the  two  objects  are  near  together. 


bQ 


112 


CHAIIV  SrRVEYIIVG, 


[part  II. 


(183)  Through  a  wood.     When  a  -wood  intervenes  between 

anj  two  given  Fi?.  119. 

points,  pre- 
venting one 
from  being 
seen  from  the 

other,  as  in  the  figure,  in  which  A  and  Z  are  the  given  points,  pro- 
ceed thus.  Hold  a  rod  at  some  point  B'  as  nearly  in  the  desired 
line  from  A  as  can  be  guessed  at,  and  as  far  from  A  as  possible. 
To  approximate  to  the  proper  direction,  an  assistant  may  be  sent  to 
the  other  end  of  the  line,  and  his  shouts  will  indicate  the  direction  ; 
or  a  gun  may  be  fired  there ;  or,  if  very  distant,  a  rocket  may  be 
sent  up  after  dark.  Then  range  out  the  "  random  line  "  AB',  by 
the  method  given  in  Art.  (169),  noting  also  the  distance  from  A 
to  each  point  found,  till  you  arrive  at  a  point  Z',  opposite  to  the 
point  Z,  i.  e.  at  that  pomt  of  the  line  from  which  a  perpendicular 
there  erected  would  strike  the  point  Z.  Measure  Z'Z.  Then 
move  each  of  the  stakes,  perpendicularly  from  the  hne  AZ',  a  dis- 
tance proportional  to  their  distances  from  A.  Thus,  if  AZ'  be 
1000  links,  and  Z'Z  be  10  links,  then  a  stake  B',  200  links  from 
A,  should  be  moved  2  links  to  a  point  B,  which  will  be  in  the  de- 
sired straight  line  AZ ;  if  C  be  400  links  from  A,  it  should  be 
moved  4  links  to  C,  and  so  with  the  rest.  The  line  should  then 
be  cleared,  and  the  accuracy  of  the  position,  of  these  stakes  tested 
by  ranging  from  A  to-  Z. 


(184)  To  an  invisible  intersection.    Let  AB  and  CD  be  two 

lines,  which,  if  prolong-  ^'^S-  120. 

ed,  would   meet  in   a  ^c: yr-&^Wzt^S?S^^z,"-J 

point  Z,  invisible  from 

either  of  them ;  and  let 

P  be  a  point,  from  which 

a  line  is  required  to  be 

set  out,  tending  to  this 

invisible      intersection. 

Set  stakes  at  the  five  given  points,  A,  B,  C,  D,  P.     Set  a  siitb 

stake  at  E,  in  the  alinements  of  AD  and  CP ;  and  a  seventh  stake 


CHAP,  v.] 


Obstacles  to  Measurement. 


113 


Then  set  an  eighth  stake 
PGr  "vnll  be  the  required 


at  F,  in  the  alinements  of  BC  and  AP. 
at  G,  in  the  alinements  of  BE  and  DF. 
line. 

Otherwise ;  Through  P  range  out  a  pai-allel  to  the  line  BD. 
Note  the  points  where  this  parallel  meets  AB  and  CD,  and  call 
these  points  Q  and  R.  Then  the  distance  from  B,  on  the  line  BD, 
to  a  point  which  shall  be  in  the  required  line  running  from  P  to  tb« 

invisible  pomt,  will  be  =  — q^ 


II.    OBSTACLES  TO  MEASUREMENT. 

(185)  The  cases,  in  which  the  direct  measurement  of  a  line  \a 
prevented  by  various  obstacles,  may  be  reduced  to  three. 

A.  When  hoili  ends  of  the  line  are  accessible, 

B.  When  one  end  of  it  is  i7iaccessible. 

C.  When  loth  ends  of  it  are  inaccessible. 


A.  WHE\  BOTH  ENDS  OF  THE  LL\E  ARE  ACCESSIBLE. 

(186)  By  perpendiculars.     On  Fig.  121. 

reaching  the  obstacle,  as  at  A  in 
the  figure,  set  off  a  perpendicular, 
AB ;  turn  a  second  right  angle  at  B, 
and  measure  past  the  obstacle ;  turn  a  third  right  angle  at  C  ;  and 
measure  to  the  original  line  at  D.  Then  will  the  measured  dis- 
tance, BC,  be  equal  to  the  desired  distance,  AD. 

If  the  direction  of  the  line  is  also  unknown,  it  will  be  most  easily 
obtained  by  the  additional  perpendiculars  shown  in  Fig.  109,  of 
Art.  (171). 


(187)  By  equilateral  triangles, 

The  method  given  m  Art.  (172),  for 
determining  the  direction  of  a  line 
through  an  obstacle,  will  also  give  its 
length ;  for  in  Fig.  121'  (Fig.  110  re- 
peated) the  desired  distance  AGis  equal 
to  the  measured  distances  AD,  or  DG. 


114 


CHAIN  SURVEYING. 


[past  n 


(188)  By  symmetrical  triangles. 

Let  AB  be  the  distance  required. 

Measure  from  A  obliquely  to  some    ^ 

point   C,  past  the  obstacle.     Mea- 

Bure  onward,  in  the  same  line,  till 

CD  is  as  long  as  AC.     Place  stakes 

at  C  and  D.     From  B  measure  to 

C,  and  from  C  measure  onward,  in 

the  same  line,  till  CE  is  equal  to  CB.     Measure  EP,  and  it  will 

be  equal  to  AB,  the  distance  required.     If  more  convenient,  make 

CD  and  CE  equal,  respectively,  to  half  of  AC  and  CB ;  then  will 

AB  be  equal  to  twice  DE. 


(189)  By  transversals.      Let 

AB  be  the  required  distance.  Set 
a  stake,  C,  in  the  line  prolonged ; ' 
set  another  stake,  D,  so  that  C  and 
B  can  be  seen  from  it ;  and  a  third 
stake,  E,  in  the  line  of  BD  pro- 
longed, and  at  a  distance  from  D 
equal  to  the  distance  from  D  to  B. 
Set  a  fourth  stake,  F,  at  the  intersection  of  EA  and  CD. 

AC 

^C,  AF  and  FE.     Then  is  AB  =  -^  (FE— AF). 


(190)  In  a  Town.  Cases  may  occur, 
m  the  streets  of  a  compactly  bmlt  town, 
in  which  it  is  impossible  to  measure  along 
any  other  luies  than  those  of  the  streets. 
The  figure  represents  such  a  case,  in 
which  is  required  the  distance,  AB,  be- 
tween points  situated  on  two  streets  which 
meet  at  the  pouit  C,  and  between  which 
runs  a  cross-street,  DE.  In  this  case 
measure  AC,  CE,  CD,  DE  and  CB. 
Then   is  the  required   distance 


Measure 


K^ 


CHAP,  v.] 


Obstacles  to  Measurement. 


115 


ACxBC) 


As  this  expression  is  somewhat  complicated,  an  example  vnW  ba 
given :  Let  AC  =  100,  CE  =  40,  CD  =  30,  DE  =  21,  and  CB 
=  80;  then  wiU  AB  =  51.7. 


B.    WUEx\  OXE  ElVD  OF  THE  LINE  IS  INACCESSIBLE. 


(191)  By  perpendiculars.  This  principle 
may  be  applied  in  a  variety  of  ways.  In  Fig. 
125,  let  AB  be  the  required  distance.  At  the 
point  A,  set  off  AC,  perpendicular  to  AB,  and  of 
any  convenient  length.  At  C,  set  off  a  perpen- 
dicular to  CB,  and  continue  it  to  a  point,  D,  in 
the  line,  of  A  and  B.  Measure  DA.  Then  is 
AC2 

"aF* 


AB  = 


Fi.cr.     125. 


(192)  Otherwise :  At  the  point  A,  in  Fig. 
126,  set  off  a  perpendicular,  AC.  At  C  set 
off  another  perpendicular,  CD.  Find  a  point, 
E,  in  the  line  of  AC,  and  BD.     Measui-e  AE 

and  EC,     Then  is  AB  =  ^^7^^. 

If  EC  be  made  equal  to  AE,  and  D  be  set 
in  the  Ime  of  BE,  and  also  in  the  perpendicular 
from  C,  then  vnU  CD  be  equal  to  AB. 

If  EC  =  i  AE,  then  CD  ==  1  AB. 

(193)  OtJienvise:  At  A,  in  Fig.  127,  mea- 
sure a  perpendicular,  AC,  to  the  line  AB  ;  and 
at  any  pomt,  as  D,  in  this  line,  set  off  a  perpen- 
dicular to  DB,  and  continue  it  to  a  point  E,  in 
the  Ime  of  CB.     Measure  DE  and  also  DA. 

AC  X  AD 


Fi>.  126 


Then  is  AB  = 


DE  — AG' 


BA 


7)  r_  /I 


Fis.  127. 


BD-/ 


116 


CHAIN  SURVEIBG. 


[part  n 


V 


(194)  By  parallels.  From  A  measure 
AC,  in  any  convenient  direction.  From  a 
point  D,  in  the  line  of  BC,  measure  a  line 
parallel  to  CA,  to  a  point  E,  in  the  line  of 
AB.     Measure  also  AE. 

AC  X  AE 


Then  is  AB 


DE  — AC" 


(195)  By  a  parallelo^^ram.  Set  a  stake,  C, 
in  the  line  of  A  and  B,  and  set  another  stake,  D, 
wherever  convenient.  With  a  distance  equal  to 
CD,  describe  from  A,  an  arc  on  the  ground ;  and, 
with  a  distance  equal  to  AC,  describe  another 
arc  from  D,  intersecting  the  first  arc  in  E.  Or,  ^  /f 
take  AC  and  CD,  so  that  together  tkey  make     i  /'' 

one  chain  ;  fix  the  ends  of  the  chain  at  A  and  D ;  o  '- 

take  hold  of  the  chain  at  such  a  link,  that  one  part  of  it  equals  AC, 
and  the  other  CD,  and  draw  it  tight  to  fix  the  point  E.  Set  a 
stake  at  F,  in  the  intersection  of  AE  and  DB.  Measure  AF  and 
AC  X  AF  ^^      AC  X  CD 


EF.     Then  is  AB 


EF 


or,  CB: 


EF 


(196)    By  syniMietrical    triang^les, 

Let  AB  be  the  required  distance.  From 
A  measure  a  line,  in  any  convenient  di- 
rection, as  AC,  and  measure  onward,  in 
the  same  direction,  till  CD  =  AC.  Take 
any  point  E  in  the  line  of  A  and  B. 
Measure  from  E  to  C,  and  onward  in  the 
same  line,  till  CF  =  CE.  Then  find  bv 
trial  a  pomt  G,  which  shall  be  at  the 
same  time  in  the  line  of  C  and  B,  and  in  G' 
the  line  of  D  and  F.  Measure  the  distance  from  G  to  D,  and  it 
will  be  equal  to  the  required  distance  from  A  to  B.  If  more  con- 
venient, make  CD  =  |  AC,  and  CF  =  ^  CE,  as  shown  by  the 
finely  dotted  lines  in  the  figure.     Then  will  DG  =  ^  AB. 


CHAP,  v.] 


Obstacles  to  Measurement. 


117 


(197)     Otherwise:   Prolong  B A  to  Fig.  I3i. 

some  point  C.  Range  out  any  con- 
venient line  CA',  and  measure  CA'  =  ^ 
CA.  The  triangle  CA'B,  is  now  to  be 
reproduced  in  a  symmetrical  triangle, 
Bituated  on  the  accessible  ground,  b 
For  this  object,  take,  on  AC,  some  point 
D,  and  measure  CD'  =  CD.  Find  the 
point  E,  at  the  intersection  of  AD'  and  A'D.  Find  the  point  F, 
at  the  intersection  of  A'B  and  CE.  Lastly,  find  the  point  B',  at 
the  intersection  of  AF  and  CA'.  Then  wHl  A'B'  =  AB.  The 
symmetrical  points  have  corresponding  letters  affixed  to  them. 


(198)  By  transversals.  Set  a  stake,  C, 
in  the  alinement  of  BA ;  a  second,  D,  at  any 
convenient  point ;  a  third,  E,  in  the  line  CD ; 
and  a  fourth,  F,  at  the  intersection  of  the 
alinements  of  DA  and  EB.  Measure  AC, 
CE,  ED,  DF  and  FA.     Then  is 

.  ^  ^ AC  X  AF  X  DE 

~  CE  X  DP  —  AF  X  DE* 

If  the  point  E  be  taken  in  the  middle  of  CD,  (as  it  is 

a        X  ,,        *  r>       AC  X  AF 
figure)  then  AB 


DF  —  AF 


If  the  point  F  be  taken  in  the  middle  of  AD,  then  AB  = 


ACx  DE 
CE  — DE' 


The  minus  signs  must  be  interpreted  as  in  Art.  (174). 


(199)  By  liariuonic  division.      Set  ^'"  ^^^ 

Brakes,  C  and  D,  on  each  side  of  A,  and 
80  that  the  three  are  in  the  same  straight 
fine.  Set  a  third  stake  at  any  point,  E, 
of  the  line  AB.  Set  a  fourth,  F,  at  the 
intersection  of  CB  and  DE  ;  and  a  fifth, 
G,  at  the  intersection  of  DB  and  CE 
Set  a  sixth  stake,  H,  at  the  intersection 

ofABandFG.    Measure  AE  and  EH.    ThenisAB  = 


AE  x  AH 
AE  —  EH' 


t 


118 


CHAIN  SrRVEYIi\G. 


[part  u 


(200)  To  an  inaccessible  line.    The  Fig.  134. 

shortest  distance,  CD,  from  a  given  point,  ^  ^ 
C,  to  an  inaccessible  straight  line  AB,  is 
required.    From  C  let  fall  a  perpendicular 
toAB,  bj  the  method  of  Art.   (158). 
Then  set  a  stake  at  any  point,  E,  on  the 
line  AC ;  set  a  second,  F,  at  the  inter- 
section of  EB  and*  CD ;  a  thu-d,  G,  at  r 
the  intersection  of  AF  and  CB  ;  and  a  fourth,  H,  at  the  interseo* 
tion  of  EG  and  CD.     Measure  CH  and  HF.     Then  is 

^-p^        CHxCF  ^j-.       ^Tj-    CH+HP  ^-p^        CHxCF 

CD  = ;  or,  CD  =  CH  . ■ or,  CD  = 

CH— HF'      '  CH— HF        '  2CH— CF 

OtJierwise  ;    When  the  inaccessible  line  is  determined  by  the 

method  of  Art.  (205)  or  (206),  the  distance  from  any  point  to  it, 

can  be  at  once  measured  to  its  symmetrical  representative. 


(201)  To  an  inaccessible  intersection.  WTien  two  lines  (as 
AB,  CD,  in  the  figure)  meet  in  a     '  Fi?.  135. 

river,  a  building,  or  any  other 
inaccessible  point,  the  distance 
from  any  point  of  either  to  their  " 
intersection,  DE,  for  example, 
may  be  found  thus.  From  any 
point  B,  on  one  Une,  measure  ^ 
BD,  and  continue  it,  till  DF  =  DB.  From  any  other  pomt,  G, 
of  the  former  line,  measure  GD,  and  continue  the  luie  till  DH  =  GD. 
Continue  HF  to  meet  DC  in  some  point  K.  Measure  KD.  KD 
will  be  equal  to  the  desired  distance  DE. 

BE  can  be  found  by  measuring  FK,  which  is  equal  to  it. 

If  DF  and  DH,  be  made  respectively  equal  to  one-half,  or  ')ne- 
tiurd,  &e.,  of  DB  and  DG,  then  will  KD  and  KF  be  respectively 
e«iual  to  one-half  or  one-third,  &c.,  of  DE  and  BE. 


CHAP,  v.] 


Obstacles  to  Measurement. 


119 


C.  WHEi\  BOTii  E.\DS  OF  THE  LIXE  ARE  INACCESSIBLE. 

(202)  By  similar  trlani^les.       Let  AB  Fi^r  i36. 

De  the  inaccessible  distance.  Set  a  stake  at 
any  convenient  point  C,  and  find  the  distan- 
ces CA  and  CB,  by  any  of  the  methods  just 
given.  Set  a  second  stake  at  any  point,  D, 
on  the  hne  CA.     Measure  a  distance,  equal 

to  ^^  .^.  ^^,  from  C,  on  the  line  CB,  to  some  point  E. 


CA 

DE.     Then  is  AB  = 


C 

Measure 


AC  X  DE 
CD      • 


If  more  convenient,  measure  CD  in  tke 
contrary  du-ection  from  the  river,  as  in  Fig. 
137,  instead  of  towards  it,  and  in  other  re- 
spects proceed  as  before. 


(203)  By  parallels.  Let  AB  be  the  in- 
accessible distance.  From  any  point,  as  C, 
range  out  a  parallel  to  AB,  as  in  Art.  (165), 
&c.  Fmd  the  distance  CA,  by  Art.  (191), 
&c.  Set  a  stake  at  the  point  E,  the  inter- 
section of  CA  and  DB,  and  measure  CE. 
CD  X  (AC  —  CE) 


Fig.   137 


Then  is  AB 


CE 


Fig.  139. 


(204)  By  a  parallelogram.    Set 

a  stake  at  any  convenient  pouit  C.   ^ 

"Set  stakes  D  and  E,  anywhere  in 

~  the  alinements  CA  and  CB.  With 
D  as  a  centre,  and  a  length  of  the 
chain  equal  to  CE,  describe  an  arc  ; 
and  with  E  as  a  centre,  and  a  length 
of  the  chain  equal  to  CD,  describe  another  arc,  intersecting  the 
fonner  one  at  F.  A  parallelogram,  CDEF,  wUl  thus  be  formed. 
Set  stakes  at  G  and  H,  where  the  alinements  DB  and  EA  inter- 
sect the  sides  of  this  parallelogram.     Measure  CD,  DF,  GF,  FH, 


120 


CHAL\  SURVEYIXG.  [paei  ii 

CD  X  DF  X  GH 


and  HG.     The  inaccessible  distance  AB  = 

CD2  X  GH 


FG  X  FH 


If  CD  =  CE,  then  AB  = 


FG  X  FH 


(205)  By  symmetrical  triangles.     Take  any  convenient  point, 
asC.     Set  stakes  at  two  other  Fig.  140. 

points,  T)  and  D',in  the  same  ^'~  '  ^^ 

Une,  and  at  equal  distances 
from  C.  Take  a  point  E,  in 
the  line  of  AD ;  measure  from 
it  to  C,  and  onward  till  CE' 
=  CE.  Take  a  pomt  F  in 
the  line  of  BD  ;  measure  from 
it  to  C,  and  onward  till  CF'  = 
CF.  Range  out  the  Hnes  AC 
and  E'D'j  and  set  a  stake  at 
their  intersection,  A'.  Range 
out  the  lines  BC  and  F'D',  and  set  a  stake  at  their  intersection, 
B'.     Measure  A'B'.     It  will  be  equal  to  the  desired  distance  AB. 


(206)  Otherwise:  Take 
any  convenient  point,  as  C, 
and  set  off  equal  distances  p 
on  each  side  of  it,  in  the 
lmeofCA,toDandD'.  Set 
off  the  same  distances  from 
C,  in  the  line  of  CB,  to  E  and 
E'.  Through  C,  set  out  a 
parallel  to  DE,  or  D'E',  and 
set  stakes  at  the  points  F 
and  F'  where  this  parallel 
intersects  AE'  and  BD'. 
Range  out  the  lines  AD'  and  EF',  and  set  a  stake  at  their  inter- 
section  A'.  Range  out  the  Hnes  BE'  and  DF,  and  set  a  stake  at 
their  intersection  B'.  Measure  A'B',  and  it  will  be  equal  to  the 
desired  distance  AB. 


CUAP.  V.J 


Obsiacles  to  Measurement. 


121 


The  easiest  method  of  settmg  out  the  parallel  in  the  above  case, 
is  to  fix  the  middle  of  the  chain  at  the  pomt  C,  and  its  ends  on  the 
hnes  CD,  CE' ;  then  carry  the  middle  of  the  chain  from  C  towards 
F,  and  mark  the  point  to  which  it  reaches ;  and  repeat  this  on  the 
other  side  of  C,  as  shown  by  the  finely  dotted  lines  m  the  figure. 

INACCESSIBLE    AREAS 

(2C7)  Triangles,  In  the  case  of  a  triangular  field,  m  which 
one  side  cannot  be  measured,  or  determined  by  any  of  the  methods 
just  given,  the  two  accessible  sides  may  be  prolonged  to  their  full 
length,  and  an  equal  symmetrical  triangle  formed,  all  of  whose  sides 
can  be  measured.  Thus  in  Fig.  102,  page  103,  if  CDE  be  the 
original  triangle,  of  which  the  side  EC  is  inaccessible,  DFP  will  be 


equal  to  it.  But  if  this  also  be  impossible,  por- 
tions of  the  sides  may  be  measui-ed,as  AD,  AE,  B 
in  the  figure  in  the  margin,  and  also  DE,  and 
the  area  of  this  triangle  found  by  any  of  the 
methods  which  have  been  given.  Then  is  the 
desu-ed  area  of  the  triangle  ABC  =  area  of 
AB  X  AC 


Fig.  142. 

\- vlfe,___ 


ADE   X 


AD  X  AE' 


(208)  Quadrilaterals.     In  the  case 

of  a  four-sided  field,  whose  sides  cannot 
be  measured^  or  prolonged,  but  whose 
diagonals  can  be  measured,  the  area 
may  be  obtained  thus.  Measure  the 
diagonals  AC  and  BDj  and  also  the 
portions  AE,  EC,  into  which  one  of 
them  is  divided  by  the  other.  Calcu- 
late the  area  of  the  triangle  BCE^by  the  preceding  method,  or  any  [^ 
of  those  heretofore  given.  Then  the  area  of  the  Quadrilateral '  ^  '"'"f  ^''^^ 
ABCD  =  areaofBCExd£^S2.     :-^rJ^'r   z.!?,^^'^-*^ 


BE  X  CE 


7'-^^ 


Methods  for  obtaming  the  areas  of  mac-  x  P/> 


(209)   Polygons. 

cessible  fields  of  more  than  four  sides,  have  been  given  in  Arts. 

OOl,)  &c.  nr.  r-.  -^-  .      —        ,-1  -  - 


FAUT  LIT. 

COMPASS    SURVEYING; 

OR 

By  the  Third  3Iethod. 


CHAPTER  I. 


A^G^LAR  SURVEYDG  L\  GENERAL. 

(210)  Angular  Surveying  determines  the  relative  positions  of 
points,  and  therefore  of  lines,  on  the  Third  principle,  as  ex- 
plained in  Art.  (7),  which  should  now  be  referred  to 

(211)  When  the  two  lines  which  form  an  angle  lie  iii  the  same 
horizontal  or  level  plane,  the  angle  is  called  a  horizontal  angle* 

When  these  lines  he  in  a  plane  perpendicular  to  the  former,  the 
angle  is  called  a  vertical  angle. 

"\Yhen  one  of  the  Hnes  is  horizontal  and  the  other  line  from  the 
eye  of  the  observer  passes  above  the  former,  and  in  the  same  ver^ 
tical  plane,  the  angle  is  called  an  angle  of  elevation. 

"WTien  the  latter  line  passes  below  the  horizontal  hne,  and  in  the 
same  vertical  plane,  the  angle  is  called  an  angle  of  depression. 

When  the  two  hnes  which  form  an  angle,  lie  in  other  planes 
which  make  oblique  angles  with  each  of  the  former  planes,  the 
angle  is  called  an  oblique  angle. 

Horizontal  angles  are  the  only  angles  employed  in  common  land 
surveying. 

•  A  plane  is  said  to  be  horizontal,  or  level,  when  it  is  pai-allel  to  the  surface  oi 
randing  water,  or  perpendicular  to  a  plumb-line.  A  line  is  horizontal  when  . 
lies  in  a  iionzonta]  plane. 


[chap.  I. 


Angular  Surveying  in  general. 


123 


(212)  The  angles  between  the  directions  of  two  lines,  which  it 
is  necessary  to  measure,  may  be  obtained  by  a  great  variety  of  in- 
struments. All  of  them  are  in  substance  mere  modifications  of  the 
very  simple  one  which  will  now  be  described,  and  wliich  any  one 
can  make  for  himself. 


(213)  Provide  a  circular  piece  of  ^■'?•  i-^^. 

wood,  and  divide  its  circumference 
(by  any  of  the  methods  of  Geometri- 
cal Drafting)  into  three  hundred  and 
sixty  equal  parts,  or  "  Degrees,"  and 
number  them  as  in  the  figure.  The 
divisions  will  be  like  those  of  a  watch 
face,  but  six  times  as  many.  These 
divisions  are  termed  graduations. 
The  figure  shows  only  every  fifteenth 
one.  In  the  centre  of  the  circle, 
fix  a  needle,  or  sharp-pointed  wire,  and  upon  this  fix  a  straight 
stick,  or  thin  ruler  placed  edge-wise,  (called  an  alidade),  so  that 
it  may  turn  freely  on  this  point  and  nearly  touch  the  graduations 
of  the  circle.  Fasten  the  circle  on  a  stafif,  pointed  at  the  other  end, 
and  long  enough  to  bring  the  aUdade  to  the  height  of  the  eyes. 
The  instrument  is  now  complete.  It  may  be  called  a  Croniometer, 
or  Ano;le-measurer. 


— i-C 


(214)  Now  let  it  be  required  to  measure  Fiir.  145. 

the  angle  between  the  lines  AB  and  AC.  Fix 
the  staff  in  the  ground,  so  that  its  centre  shall 
be  exactly  over  the  intersection  of  the  two 
hnes.  Turn  the  alidade,  so  that  it  points,  (as 
determined  by  sighting  along  it)  to  a  rod,  or 
other  mark  at  B,  a  point  on  one  of  the  hnes,  and  note  what  degree 
it  covers,  i.  e.  "  The  Reading."  Then,  without, disturbing  the 
circle,  turn  the  alidade  till  it  points  to  C,  a  point  on  the  other  line. 
Note  tlie  new  reading.  The  difference  of  these  readmgs,  (in  the 
figure,  45  degrees),  is  che  difference  in  the  directions  of  the  two 
lines,  or  is  the  angle  which  one  makes  with  the  other.     If  the  dia 


124  COMPASS  SlRVEl L\G.  [part  m 

tance  from  A  to  C  be  now  measured,  the  point  C  is  "  determined,'' 
with  respect  to  the  points  A  and  B,  on  the  Third  Princijjle.  An^ 
number  of  points  may  be  thus  determined. 

(215)  Instead  of  the  verj  simple  and  rude  alidade,  which  has 
been  supposed  to  be  used,  needles  may  be  fixed  on  each  end  of  the 
aUdade  ;  or  sights  may  be  added,  such  as  those  described  in  Art. 
(106)  ;  or  a  small  straight  tube  may  be  used,  one  end  bemg  cover- 
ed with  a  piece  of  pasteboard  in  which  a  very  small  eye  hole  is 
pierced,  and  threads,  called  "  cross-hairs,"  bemg  stretch-  ^"-s  i^G. 
ed  across  the  other  end  of  it,  as  in  the  figure  ;  so  that  (^  (^ 
their  mtersection  may  give  a  more  precise  line  for  determining  the 
direction  of  any  point. 

(216)  "WTien  a  telescope  is  substituted  for  this  tube,  and  sup- 
ported in  such  a  way  that  it  can  turn  over,  so  as  to  look  both  back- 
wards and  forwards,  the  instrument  (with  various  other  additions, 
which  however  do  not  affect  the  principle),  is  called  the  Engineer's 
Transit. 

With  the  addition  of  a  level,  and  a  vertical  circle,  for  measm'ing 
vertical  angles,  the  instrument  becomes  a  Theodolite ;  in  which, 
however,  the  telescope  does  not  usually  admit  of  being  turned  over. 

(217)  The  Compass  differs  from  the  instruments  which  have 
been  described,  m  the  following  respect.  They  all  measure  the 
angle  which  one  line  makes  mth  another.  The  compass  measures 
the  angle  which  each  of  these  lines  makes  with  a  third  hne,  viz : 
that  shown  by  the  magnetic  needle,  which  always  points  (approxi* 
mately)  in  the  same  direction,  i.  e.  North  and  South,         f'ig-  147. 

in  the  3Iagnetic  Meridian.  Thus,  in  the  figm-e,  the  -^ 
line  AB  makes  an  angle  of  30  degrees  with  the  line 
AN,  and  the  Ime  AC  makes  an  angle  of  75  de- 
grees with  AN.  The  difference  of  these  angles, 
or  45  degrees,  is  the  angle  which  AC  makes 
with  AB,  agi-eeing  with  the  result  obtained  in 

Art  (214). 

S 


-'  C 


[chap.  1.  Angular  Survcyini?  in  f^cneral.  125 

(218)  Surveying  witli  the  compass  is,  therefore,  a  less  direct 
operation  than  surveying  with  the  Transit  or  Theodolite.  But  as 
the  use  of  the  compass  is  much  more  rapid  and  easy  (only  one  sight 
and  reading  at  each  station  heing  necessary,  instead  of  two,  as  in 
tlie  former  case),  for  this  reason,  ae  well  as  for  its  smaller  cost,  it 
is  the  instrument  most  commonly  employed  in  land  surveying  in 
this  country,  in  spite  of  its  imperfections  and  inaccuracies. 

As  many  may  wish  to  learn  "  Surveying  with  the  Compass," 
without  being  obliged  to  previously  learn  "  Surveying  with  the 
Transit,"  (which  properly,  being  more  simple  in  principle,  though 
less  so  in  practice,  should  precede  it,  but  which  will  be  considered 
in  Part  IV),  we  will  first  take  up  Compass  Surveying. 

(219)  Angular  Surveying  m  general,  and  therefore  Compass 
Surveying,  may  employ  either  of  the  3d,  4th  and  5th  methods  of 
determining  the  position  of  a  point,  given  in  Part  I ;  that  is,  any 
instrument  which  measures  angles  may  be  employed  for  Polar, 
Triangular,  or  Trilinear  Surveying.  The  first  of  these,  Polar 
Surveying,  is  the  one  most  commonly  adopted  for  the  compass,  and 
is  therefore  the  one  which  will  be  specially  explained  in  this  part. 

The  same  method,  as  employed  with  the  Transit  and  Theodolite, 
will  be  explained  in  the  following  part. 

The  4th  and  5th  methods  will  be  explained  in  the  next  two  parts. 

(220)  The  method  of  Polar  Surveying  embraces  two  minor 
methods.  The  most  usual  one  consists  in  going  around  the  field 
with  the  instrument,  setting  it  at  each  corner  and  measuring  there 
the  angle  which  each  side  makes  with  its  neighbor,  as  well  as  the 
length  of  each  side.  This  method  is  called  by  the  French  the  me- 
thod of  Cheminement.  It  has  no  special  name  in  English,  but  may 
be  called  (from  the  American  verb.  To  progress),  the  Method  of 
Progression.  The  other  system,  the  3IetIiod  of  Radiation,  con- 
sists m  setting  the  instrument  at  one  point,  and  thence  measuring 
the  du-ection  and  distance  of  each  corner  of  the  field,  or  other 
object.  The  corresponding  name  of  what  we  have  called  Triangu- 
lar Surveying  is  the  Method  of  Intersections  ;  smce  it  determines 
points  by  the  intersections  of  straight  lines. 


126 


COMPASS  SURVEYING. 


fPART   III. 


O  (^^3^ 


CHAPTER  11. 

THE  COMPASS. 

(221)  Til*  Needle.  The  most  essential  part  of  the  compass  ia 
the  magnetic  needle.  It  is  a  slender  bar  of  steel,  usually  five  or 
Bix  inches  long,  strongly  magnetized,  and  balanced  on  a  pivot,  sc 
that  it  may  turn  freely,  and  thus  be  enabled  to  continue  pointing 
in  the  same  direction  (that  of  the  "  Magnetic  3Ieridian,^^  approxi- 
mately North  and  South)  however  much  the  "  Compass  Box,"  to 
which  the  pivot  is  attached,  may  be  turned  around. 

As  it  is  important  that  the  needle  should  move  with  the  least 
possible  friction,  the  pivot  should  be  of  the  hardest  steel  ground  to 
a  very  sharp  point ;  and  in  the  centre  of  the  needle,  which  is  to 
rest  on  the  pivot,  should  be  inserted  a  cap  of  agate,  or  other  hard 
material.  Iridium  for  the  pivot,  and  ruby  for  the  cap,  are  still 
better. 

If  the  needle  be  balanced  on  its  pivot  before  being  magnetized, 
one  end  will  sink,  or  "  Dip,"  after  the  needle  is  magnetized.  To 
biing  it  to  a  level,  several  coils  of  wire  are  wound  around  the  nee- 
dle so  that  they  can  be  sUd  along  it,  to  adjust  the  weight  of  its  two 
ends  and  balance  it  more  perfectly. 

The  North  end  of  the  needle  is  usually  cut  into  a  more  orna- 
mental form  than  the  South  end,  for  the  sake  of  distmction. 

The  principal  requisites  of  a  compass  needle  are,  intensity  of  di- 
rective force  and  susceptibility.     "Shear  steel"  was  found  by 
Capt.  Kater  to  be  the  kind  capable  of  receiving  the  greatest  mag- 
netic force.     The  best  form  is  that  of  a  rhomboid.         Fig.  149. 
or  lozenge,  cut  out  in  the  middle,  so  as  to  dimi- 


nish the  extent  of  surface  in  proportion  to  the 
mass,  as  it  is  the  latter  on  which  the  du^ective  force  depends.  Ee- 
yond  a  certain  limit,  say  five  inches,  no  additional  power  is  gained 
by  increasing  the  length  of  the  needle.  On  the  contrary,  longer 
ones  are  apt  to  have  their  strength  diminished  by  several  consecu- 
tive poles  being  formed.  Short  needles,  made  very  hard,  are 
therefore  to  be  preferred. 


128  COMPASS  SURVEYING.  [part  hi 

The  needle  should  not  come  to  rest  very  quicklj.  If  it  does,  it 
indicates  either  that  it  is  weakly  magnetized,  or  that  the  friction  on 
the  pivot  is  great.  Its  sensitiveness  is  indicated  by  the  number  of 
vibrations  which  it  makes  in  a  small  space  before  coming  to  rest. 

A  screw,  with  a  milled  head,  on  the  under  side  of  the  plate 
which  supports  the  pivot,  is  used  to  raise  the  needle  off  this  pivot, 
when  the  instrument  is  carried  about,  to  prevent  the  point  being 
dulled  by  unnecessary  friction. 

(222)  The  Sights.  Next  after  the  needle,  which  gives  the  di 
rection  of  the  fixed  line,  whose  angles  with  the  lines  to  be  survey- 
ed are  to  be  measured,  should  be  noticed  the  Sights,  which  show 
the  directions  of  these  last  lines.  At  each  end  of  a  line  passmg 
through  the  pivot  is  placed  a  "  Sight,"  consisting  of  an  upright  bar 
of  brass,  with  openings  in  it  of  various  forms ;  usually  either  slits, 
with  a  circular  aperture  at  their  top  and  bottom* ;  or  of  the  form 
described  in  Art.  (106) ;  all  these  arrangements  being  intended  to 
C-iable  the  Une  of  sight  to  be  directed  to  any  desired  object,  with 
precision. 

(223)  A  Telescope  which  can  move  up  and  down  in  a  vertical 
plane,  i.  e.  a  plunging  telescope,  or  one  which  can  turn  completely 
over,  is  sometimes  substituted  for  the  sights.  It  has  the  great 
advantage  of  giving  more  distinct  vision  at  long  distances,  and  of 
admitting  of  sights  up  and  down  very  steep  slopes.  Its  accuracy 
of  vision  is  however  rendered  nugatory  by  the  want  of  precision  in 
the  readings  of  the  needle.  If  a  telescope  be  applied  to  the  com 
pass,  a  graduated  circle  watli  vernier  should  be  added,  thus  con- 
verting the  compass  into  a  "  Transit."  The  Telescope  wiU  be 
found  mmutely  described  in  Part  IV,  "  Transit  Surveying." 

V 

C224)  The  divided  circle.  We  now  have  the  means  of  indi- 
cating the  directions  of  the  two  lines  whose  angle  is  to  be  measur- 
ed. The  number  of  degrees  contained  in  it  is  to  be  read  from  a 
circle,  divided  into  degrees,  in  the  centre  of  which  is  fixed  the 

*  An  inside  and  an  outside  view,  or  "  Elevation,"  of  such  sights,  are  given  on 
each  side  of  the  figure  of  the  Compass,  on  page  126.  It  is  itself  drawn  in  "  Mili 
tary  Perspective." 


OHAP.  II.] 


The  Compass. 


129 


pivot  bearing  the  needle.  The  graduations  are  usually  made  to 
half  a  degree,  and  a  quarter  of  a  degree  or  less  can  then  be  "  esti- 
mated." The  pivot  and  needle  are  smik  in  a  circular  box,  so  that 
its  top  maj  be  on  a  level  with  the  needle.  The  graduations  are 
usually  made  on  the  top  of  the  surrounding  rim  of  the  box,  but 
should  also  be  continued  down  its  inside  circumference  so  that  it  may 
be  easier  to  see  with  what  division  the  ends  of  the  needle  coincide. 

The  degrees  are  not  numbered  consecutively  from  0°  around  to 
360^  ;  but  run  from  0^  to  90°,  both  ways  from  the  two  diametii- 
cally  opposite  pouats  at  which  a  Une,  passing  through  the  shts  in  the 
middle  of  the  sights,  would  meet  the  divided  circle. 

The  lettering  of  the  Surveyor's  Compass  has  one  important  dif 
ference  from  that  of  the  Mariner's  Compass. 

When  we  stand  facing  the  North,  the  East  is  on  our  right  hand, 
and  the  West  on  our  left.  The  graduated  card  of  the  Mariner's 
Compass  which  is  fastened  to  the  needle,  and-  turns  with  it,  is 
marked  accordingly.  But,  in  the  Surveyor's  compass,  one  of  the 
0  points  being  marked  N,  or  North,  (or  indicated  by  a  fleur-de- 
lis,)  and  the  opposite  one  S,  or  South,  the  90-degrees-point  on  the 
right  of  this  line,  as  you  stand  at  the  S  end  and  look  towards  the 
N,  is  marked  W,  or  West ;  and  the  left  hand  90-degrees-point  iai 
marked  E,  or  East.  The  reason  of  this  will  be  seen  when  the 
method  of  using  the  compass  comes  to  be  explained  in  the  following 
chapter. 


(225)  The  Points.  In  or- 
dinary land  surveying,  only  four 
points  of  the  compass  have 
names,  viz :  North,  South,  East 
and  West;  the  direction  of  a 
line  being  described  by  the  an- 
gle which  it  makes  wth  a  North 
and  South  line,  to  its  East  or  to 
its  West.  But  for  nautical  pur- 
poses, the  circle  of  the  compass 
is  divided  into  32  points,  the 
names  of  which  are  shown  in 


130 


CO^IPASS  SURVEYING. 


[part  iil 


\h.e  figure.  Two  rules  embrace  all  the  cases.  1°  When  the 
letters  indicating  two  points  are  joined  together,  the  point  half  way 
between  the  two  is  meant ;  thus.  N.  E.  is  half  way  between  North 
and  East ;  and  N.  N.  E.  is  half  way  between  North  and  North 
East.  2°  When  the  letters  of  two  points  are  joined  together 
with  the  intermediate  word  b^,  it  indicates  the  point  which  comes 
next  after  the  first,  in  going  towards  the  second  ;  thus,  N.  by  E,  is 
the  point  which  follows  North  in  going  towards  the  East ;  8.E.  by 
S.  is  the  next  point  from  South  East,  going  towards  the  South. 

(226)  Eccentricity.  The  centre-pin,  or  pivot  of  the  needle, 
ought  to  be  exactly  in  the  centre  of  the  graduated  circle  ;  the  nee- 
dle ought  to  be  straight ;  and  the  line  of  the  sights  ought  to  pass 
exactly  through  this  centre  and  through  the  0  points  of  the  circle. 
If  this  is  not  the  case,  there  will  be  an  error  in  every  observation. 
This  is  called  the  error  of  eccentricity/. 

When  the  maker  of  a  compass  is  about  to  fix  the  pivot  in  place, 
he  is  in  doubt  of  two  things  ;  whether  the  needle  is  perfectly  straight, 
and  whether  the  pivot  is  exactly  in  the  cen- 
tre. In  figures  151  and  152,  both  of  these 
are  represented  as  being  excessively  in 
error. 

Firstly,  to  examine  if  the  needle  be 
straight.     Fix  the  pivot  temporarily,   so 
that  the  ends  of  the  needle  may  cut  oppo- 
site degrees,  i.  e.   degrees   differing  by 
180^.     The  condition  of  things  at   this 
stage  of  progress,  will  be  represented  by 
Fig.   151.     Then  turn  the  compass-box 
half  way  around.     The  error  will  now  be 
doubled,  as  is  shown  by  Fig.  152,  in  which 
the  former  position  of  the  needle  is  indi- 
cate! by  a  dotted  line.*     Now  bend  the 
needle,  as  in  Fig.  153,  till  it  cuts  divi- 
sions midway  between  those  cut  by  it  in 


*  This  is  another  example  of  the  fruitful  priuci  jle  of  Reversion,  first  noticed  la 
Art.  (105;. 


CHAP     II.] 


The  Compass. 


131 


its  present  and   in  its  former  position 
This  makes  it  certain  that  the  needle  is 
straight,  or  that  its  two  ends  and  its  cen- 
ti'e  lie  in  the  same  straight  line. 

Secondly,  to  put  the  pivot  in  the  cen- 
tre. ]\Iove  it  till  the  straightened  needle 
cuts  opposite  di^dsions.  It  is  then  certain 
that  the  direction  of  the  needle  passes 
tJirough  the  centre.  Turn  the  compass 
box  one-quarter  around,  and  if  the  needle  does  not  then  cut  oppo- 
site divisions,  move  the  pivot  till  it  does.  Repeat  the  operation  in 
various  positions  of  the  box.  It  will  be  a  sufficient  test  if  it  cuts 
the  opposite  divisions  of  0^,  45°  and  90°. 

To  fix  the  sights  precisely  in  hue,  draw  a  hair  through  their  slits 
and  move  them  till  the  hair  passes  over  the  0  points  on  the  circle. 

The  surveyor  can  also  examine  for  himself,  by  the  principle  of 
Keversion,  whether  the  hne  of  the  sights  passes  through  the  centre 
or  not.  Sight  to  any  very  near  object.  Read  oflf  the  number  of 
degrees  mdicated  by  one  end  of  the  needle.  Then  turn  the  com- 
pass half  around,  and  sight  to  the  same  object.  If  the  two  read- 
ings do  not  agree,  there  is  an  error  of  eccentricity,  and  the  arith- 
metical mean,  or  half  sum  of  the  two  readings  is  the  correct  one. 

Fi-.  154.  Fig.  15.5. 


In  Fig.  154,  the  hne  of  sight  AB  is  represented  as  passing  to 
one  side  of  the  centre,  and  the  needle  as  pointing  to  46°.  In  Fig. 
155,  the  compass  is  supposed  to  have  been  turned  half  around  and 
the  other  end  of  the  sights  to  be  directed  to  the  same  object. 
Suppose  that  the  needle  would  have  pointed  to  45°,  if  the  hne  of 


132  COMPASS  SIMEYL\G.  [part  m 

Bight  Lad  passed  through  the  centre.  The  needle  will  now  poini 
to  44*^,  the  error  being  doubled  bj  the  reversion,  and  the  true 
reading  being  the  mean. 

This  does  not,  however,  make  it  certain  that  the  line  of  the 
sights  passes  through  the  0  points,  which  can  only  be  tested  by  the 
hair,  as  mentioned  above. 

(227)  Levels.  On  the  compass  plate  are  two  small  spirit  levels. 
They  consist  of  glass  tubes,  shghtly  curved  upwards,  and  nearly 
filled  with  alcohol,  leavmg  a  bubble  of  air  within  them.  They 
are  so  adjusted  that  when  the  bubbles  are  in  the  centres  of  the 
tubes,  the  plate  of  the  compass  shall  be  level.  One  of  them  lies  in 
the  direction  of  the  sights,  and  the  ether  at  right  angles  to  this 
direction. 

(228)  Tana^ent  Scale.  This  is  a  convenient,  though  not  essen- 
tial, addition  to  the  compass,  for  the  purpose  of  measuring  the 
slopes  of  ground,  so  that  the  proper  allowance  in  chaining  may  be 
made.  In  the  figure  of  the  compass,  page  126,  may  be  seen,  on 
the  edge  of  the  left  hand  sight,  a  small  projection  of  brass  with  a 
hole  through  it.  On  the  edge  of  the  other  sight  are  engraved 
lines  numbered  from  0°  to  20°,  the  0°  being  of  the  same  height 
ft,bove  the  compass  plate  that  the  eye-hole  is.  To  use  this,  set  the 
compass  at  the  bottom  of  a  slope,  and  at  the  top  set  a  signal  of 
exactly  the  height  of  the  eye-hole  from  the  ground.  Level  the 
compass  very  carefully,  particularly  by  the  level  which  hes  length- 
mse,  and,  with  the  eye  at  the  eye-hole,  look  to  the  signal  and  note 
the  number  of  the  division  on  the  farther  sight  which  is  cut  by  the 
visual  ray.  That  will  be  the  angle  of  the  slope  ;  the  distances  of 
the  engraved  lines  from  the  0°  line  being  tangents  (for  the  radius 
equal  to  the  distance  between  the  sights)  of  the  angles  correspond- 
ing to  the  numbers  of  the  lines. 

(229)  Vernier,  The  compass  box  is  connected  with  the  plate, 
wliich  carries  it  and  the  sights,  so  that  it  can  turn  around  on  thi3 
plate.  This  motion  is  given  to  it  by  a  screw,  (called  a  slow-mo- 
tion, or  Tangent  screw),  the  head  of  which  is  the  nearest  one  m 


CHAP    I.] 


The  Compass. 


133 


the  figure  on  page  12G.  K  two  marks  be  made  opposite  to  each 
other,  one  on  the  projecting  part  of  the  compass  box,  and  the  other 
on  the  plate  to  which  the  sights  are  fastened,  these  marks  will  separ 
rate  when  the  slow-motion  screw  is  turned.  Their  distance  apart 
(in  angular  measurement,  i.  e.  fractions  of  a  circle),  in  any  posi- 
tion, is  measured  bj  a  contrivance  called  a  Vernier,  which  is  the 
minutely  di\dded  arc  of  a  circle  seen  between  the  left  hand  sight 
and  the  compass  box.  It  will  be  better  to  defer  explaining  the 
mode  of  reading  the  vernier  for  the  present,  since  it  is  rarely  used 
with  the  compass,  and  an  entire  chapter  will  be  given  to  it  in  Part 
IV.  Its  principle  is  similar  to  that  of  the  Vernier  Scale,  described 
m  Art.  (50).  Its  applications  in  "  Field-work "  will  be  noticed 
under  that  head. 


Fisr.  157. 


(230)  Tripod.  The  compass,  hke  most  surveying  instruments, 
is  usually  supported  on  a  Tripod,  consisting  of  three  legs,  shod  with 
iron,  and  so  connected  at  top  as  to  be  movable  in  any  direction. 
There  are  many  forms 
of  these.  Lightness 
and  stiffness  are  the 
qualities  desired.  The 
most  usual  form  is 
shewn  in  the  figures 
of  the  Transit  and  the 
Theodohte  at  the  be- 
ginning of  Part  IV. 
Of  the  two  represent- 
ed in  Figs.  156  and 
157,  the  first  has  the 
advantage  of  being  ve- 
ry easily  and  cheaply 
made  ;  and  the  second 
that  of  beino;  fio'ht  and 
yet  capable  of  very  firmly  resisting  horizontal  torsion. 

The  joints,  by  which  the  instrument  is  connected  with  the  tripod, 
are  also  various.  Fig.  158  is  the  "  Ball-and-socket  jomt,"  most 
usual  in  this  country.     It  takes  its  name  from  the  ball,  in  which 


134  i'OMPASS  SURVEYING.  [fart  hi 

Fi^'.  153.  Fig.  159.  Fig.  160 


terminates  tlie  covered  spindle  which  enters  a  corresponding  cavity 
under  the  compass  plate,  and  the  socket  in  which  this  ball  turns. 
It  admits  of  motion  in  any  direction,  and  can  be  tightened  or  loos- 
ened by  turning  the  upper  half  of  the  hollow  piece  enclosing  it, 
which  is  screwed  on  the  lower  half.  Fig.  159  is  called  the  "  Shell- 
joint."  In  it  the  two  shell-shaped  pieces  enclosing  the  ball  are 
tightened  by  a  thumb-screw.  Fig.  160,  is  "  Cugnot's  joint."  It 
consists  of  two  cyluiders,  placed  at  right  angles  to  each  other,  and 
through  the  axes  of  which  pass  bolts,  which  turn  freely  in  the  cylin- 
der and  can  be  tightened  or  loosened  by  thumb-screws  at  their 
ends.  The  combination  of  the  two  motions  which  this  joint  permits, 
enables  the  instrument  which  it  carries,  to  be  placed  ui  any  desired 
position.     This  joint  is  much  the  most  stable  of  the  three. 

(231)  Jacob's  Staff.  A  smgle  leg,  called  a  "  Jacob's  Staff," 
has  some  advantages,  as  it  is  lighter  to  carry  in  the  field,  and  can 
be  made  of  any  wood  on  the  spot  where  it  is  to  be  used,  thus  sav 
ing  the  expense  of  a  tripod  and  the  trouble  of  its  transportation 
Its  upper  end  is  fitted  into  the  lower  end  of  a  brass  head  which  has 
a  ball  and  socket  joint,  and  axis  above.  Its  lower  end  should  be 
shod  with  iron,  and  a  spike  running  through  it  is  useful  for  pressing 
it  into  the  ground  with  the  foot.  Of  course  it  cannot  be  conven 
iently  used  on  frozen  ground,  or  on  pavements.  It  may,  however, 
be  set  before  or  beliind  the  spot  at  which  the  angle  is  to  be  mea- 


CHAP.  II. J 


The  Compass. 


135 


sured,  provided  that  it  is  placed  very  precisely  iii  the  line  of  direo 
tion  from  that  station  to  the  one  to  which  a  sight  is  to  be  t^iken. 


(232)  The  Prismatic  Compass.  The  peculiarity  of  this  instru- 
ment (often  called  Schmalcalder's)  is  that  a  glass  triangular  prism 
is  substituted  for  one  of  the  sights.  Such  a  prism  has  this  peculiar 
property  that/at  the  same  time)  it  can  be  seen  through,  so  that  a 


sight  can  be  taken  through  it,  and  that  its  upper  surface  reflects 
Uke  a  mirror,  so  that  the  numbers  of  the  degrees  immediately  under 
it,  can  be  read  off  at  the  same  time  that  a  sight  to  any  object  is 
taken.  Another  peculiarity,  necessary  for  profitmg  by  the  last 
one,  is,  that  the  divided  circle  is  not  fixed,  but  is  a  card  fastened 
to  the  needle  and  moving  around  with  it,  as  in  the  Mariner's  Com- 
pass. The  minute  description,  which  follows,  is  condensed  from 
Simms. 

In  the  figure,  A  repre- 
sents the  compass  box,  and 
B  the  card,  which,  being 
attached  to  the  magnetic 
needle,  moves  as  it  moves, 
around  the  agate  centre, 
a,  on  which  it  is  suspend- 
ed. The  circumference 
of  the  card  is  usually  di- 
vided to  I  or  I  of  a  de- 
gree. C  is  a  prism,  which 
the  observer  looks  through. 
The  perpendicular  thread 
of  the  sight- vane,  E,  and 
the  divisions  on  the  card,  appear  together  on  looking  through  the 
prism,  and  the  division  with  which  the  thread  coincides,  when  the 
needle  is  at  rest,  is  the  "  Bearing"  of  whatever  object  the  thread 
may  bisect,  i.  e.  is  the  angle  which  the  line  of  sight  makes  with  the 
direction  of  the  needle.  The  prism  is  mounted  with  a  hinge  joinc, 
D.  The  sight-vane  has  a  fine  thread  stretched  along  its  opening, 
in  the  direction  of  its  length,  which  is  brought  to  bisect  any  object, 
by  turning  the  box  around  horizontally.     F  is  a  mirror,  made  to 


136 


COMPASS  SURVEYING. 


[PAKT  Itl 


slide  on  or  off  the  siglit-vane,  E  ;  and  it  may  be  reversed  at  pie* 
Bure,  that  is,  tui-ned  face  downwards ;  it  can  also  be  inclined  at 
any  angle,  by  means  of  its  joint,  d ;  and  it  will  remain  stationary 
on  any  part  of  the  vane,  by  the  friction  of  its  slides.  Its  use  is  t<? 
reflect  the  image  of  an  object  to  the  eye  of  an  observer  when  the 
object  is  much  above  or  below  the  horizontal  plane.  The  colored 
glasses  represented  at  G,  are  intended  for  observing  the  sun.  At 
e,  is  shown  a  spring,  which  being  pressed  by  the  finger  at  the  time 
of  observation,  and  then  released,  checks  the  vibrations  of  the  card, 
and  brings  it  more  speedily  to  rest.  A  stop  is  likewise  fixed  to 
the  other  side  of  the  box,  by  which  the  needle  may  be  thrown  oflf 
its  centre. 

The  method  of -using  this  instrument  is  very  simple.  First  raise 
the  prism  in  its  socket,  5,  untU  you  obtain  a  distinct  view  of  the 
divisions  on  the  card.  Then,  standing  over  the  point  where  the 
angles  are  to  be  taken,  hold  the  instrument  to  the  eye,  and,  looking 
through  the  sht,  (7,  turn  around  tiU  the  thread  in  the  sight-vane 
bisects  one  of  the  objects  whose  beaiing  is  required  ;  then  by  touch- 
ing the  spring,  e,  bring  the  needle  to  rest,  and  the  di\'ision  on  the 
card  which  coincides  with  the  thread  on  the  vane,  will  be  the  bear- 
ing of  the  object  from  the  north  or  south  points  of  the  magnetic 
meridian.  Then  turn  to  any  other  object,  and  repeat  the  opera- 
tion ;  the  difference  between  the  bearing  of  this  object  and  that  of 
the  former,  will  be  the  angular  distance  of  the  objects  in  question. 
Thus,  suppose  the  former  bearing  to  be  40^  30',  and  the  latter 
10^  15',  both  east,  or  both  west, 
from  the  noHh  or  south,  the  angle 
wiU  be  30^  16'.  The  divisions  are 
generally  numbered  5°,  10°,  15'°, 
&c.  around  the  circle  to  360°. 

The  figures  on  the  compass  card 
are  le versed,  or  written  upside 
down,  as  in  the  figure  (in  which 
only  every  fifteenth  degree  is  mark- 
ed), because  they  are  again  re- 
rersed  by  the  prism. 


Fig.  162. 


CHAP  II.]  The  Compass.  137 

(233)  The  prismatic  compass  is  generally  held  in  the  hand,  the 
bearing  being  caught,  as  it  were,  in  passing ;  but  more  accurate 
readings  would  of  course  be  obtained  if  it  rested  on  a  support,  such 
as  a  stake  cut  flat  on  its  top. 

In  the  former  mode,  the  needle  never  comes  completely  to  rest, 
particularly  in  the  wind.  In  such  cases,  observe  the  extreme  di- 
visions between  which  the  needle  vibrates,  and  take  their  arith- 
metical mean. 

(234)  Defects  of  compass.  Tlie  compass  is  deficient  m  both 
precision  and  correctness.* 

The  former  defect  arises  from  the  mdefiniteness  of  its  mode  of 
indicating  the  part  of  the  circle  to  wliich  it  points.  The  point  of 
the  needle  has  considerable  thickness ;  it  cannot  quite  touch  the 
di\aded  circle ;  and  these  divisions  are  made  only  to  whole  or  half 
degrees,  though  a  fraction  of  a  di\asion  may  be  estimated,  or  guessed 
at.  The  Vernier  does  not  much  better  this,  as  we  shaU  see  when 
explaining  its  use.  Now  an  inaccuracy  of  one  quarter  of  a  degree 
in  an  angle,  i.  e.  in  the  difference  of  the  directions  of  two  hnes, 
causes  them  to  separate  from  each  other  5|  inches  at  the  end  of 
100  feet ;  at  the  end  of  1000  feet  nearly  4^  feet ;  and  at  the  end 
of  a  mile,  23  feet.  A  difference  of  only  one-tenth  of  a  degree,  or 
six  minutes,  would  produce  a  difference  of  1|  feet  at  the  end  of 
1000  feet ;  and  9|  feet  at  the  distance  of  a  mile.  Such  are  the 
differences  which  may  result  from  the  want  of  precision  in  the  in- 
dications of  the  compass. 

But  a  more  serious  defect  is  the  want  of  correctness  in  the  com- 
j:<ass.  Its  not  pointing  exactly  to  the  true  north  does  not  indeed 
affect  the  correctness  of  the  angles  measured  by  it.  But  it  does  not 
point  in  the  same  or  in  a  parallel  direction,  during  even  the  same 
day,  but  changes  its  direction  between  sunrise  and  noon  nearly  a 
quai'ter  of  a  degree,  as  will  be  fully  exj^lained  in  Chapter  VIII. 
The  effect  of  such  a  difference  we  have  jast  seen.     This  direction 

*  The  student  must  not  confduiid  these  two  qualities.  To  say  that  me  snn  af>- 
pears  to  rise  in  the  eastern  quarter  of  the  heavens  and  to  set  in  the  western,  :a 
eorrect,  but  not  precise.  A  watch  with  a  second  hand  indicates  the  time  of  day 
^ecisely,  hut  not  always  correctly.  The  statement  that  two  and  two  make  five^ 
ie  precise,  but  is  not  usually  regarded  as  correct. 


138  COMPASS  SURVEIING.  [part  m 

may  also  be  greatly  altered  in  a  moment,  without  the  knowledge 
of  the  surveyor,  by  a  piece  of  iron  being  brought  near  to  the  com 
pass,  or  by  some  other  local  attraction,  as  will  be  noticed  hereafter. 
This  is  the  weak  point  in  the  compass. 

Notwithstanding  these  defects,  the  compass  is  a  very  valuable 
instrument,  from  its  simphcity,  rapidity  and  convenience  in  use ; 
and  though  never  precise,  and  seldom  correct,  it  is  generally  not 
very  wrong. 


CHAPTER  HI. 

THE  FIELD  WORK. 

(235)  Taking  Bearings.  The  "Bearing"  of  a  line  is  the  an- 
gle which  it  makes  with  the  direction  of  the  needle.  Thus,  in  Fig. 
147,  page  124,  the  angle  NAB  is  the  Bearing  of  the  Hne  AB,  and 
NAC  is  the  Bearing  of  AC.  The  Bearing  and  length  of  a  line  are 
named  collectively  the  Course, 

To  take  the  Bearing  of  any  line,  set  the  compass  exactly  over 
any  point  of  it  by  a  plumb4ine  suspended  from  beneath  the  cen- 
tre of  the  compass,  or,  approximately,  by  dropping  a  stone.  Level 
the  compass  by  bringing  the  air  bubbles  to  the  middle  of  the  level 
tubes.  Direct  the  sights  to  a  rod  held  truly  vertical,  or  "  plumb," 
at  another  point  of  the  hne,  the  more  distant  the  better.  The  two 
ends  are  usually  taken.  Sight  to  the  lowest  visible  point  of  the 
rod.  When  the  needle  comes  to  rest,  note  what  di^dsion  on  the 
circle  it  points  to  ;  taking  the  one  indicated  by  the  North  end  of 
the  needle,  if  the  North  point  on  the  circle  is  farthest  from  you, 
and  vice  versa. 

In  reading  the  division  to  w\ich  one  end  of  the  needle  points, 
the  eye  should  be  placed  over  the  other  end,  to  avoid  the  error 
which  might  result  from  the  "  parallax,"  or  apparent  change  of 
place,  of  the  end  read  from,  when  looked  at  obliquelv. 


CHAP.  III.] 


The  Field  Work. 


139 


Tlie  bearing  is  read  and  recorded  by  noting  between  what  letters 
the  end  of  the  needle  comes,  and  to  what  number ;  naming,  or 
writing  down,  firstly,  that  letter,  N  or  S,  which  is  at  the  0^  point 
nearest  to  that  end  of  the  needle  from  which  jou  are  reading ; 
secondly,  the  number  of  degrees  to  which  it  points,  and  tldrdly^ 
the  letter,  E  or  W,  of  the  90°  pomt  which  is  nearest  to  the  same 
end  of  the  needle.  Thus,  in  the  figm-e,  if  when  the  sights  W3re 
du-ected  along  a  hne,  (the  North 
point  of  the  compass  being  most 
distant  from  the  observer),  the 
North  end  of  the  needle  was  at  the 
point  A,  the  bearing  of  the  line 
sighted  on,  would  be  North  45'° 
East ;  if  the  end  of  the  needle  was 
at  B,  the  bearing  would  be  East ;  if 
at  C,  S.  30°  E  ;  if  at  D,  South;  if 
at  E,  S.  60°  W ;  if  at  F,  West ;  if 
at  G,  N.  60°  W ;  if  at  H,  North. 


164. 


— -B 


(236)  We  can  now  understand  why  W  is  en  the  right  hand  of 
the  compass-box,  and  E  on  the  left.  Let  the  direction  from  the 
centre  of  the  compass  to  the  point 
B  in  the  figure,  be  required,  and 
suppose  the  sights  in  the  first  place 
to  be  pointing  in  the  direction  of  the 
needle,  S  N,  and  the  North  sight 
to  be  ahead.  Wlien  the  sights  (and 
the  circle  to  which  they  are  fasten- 
ed) have  been  turned  so  as  to  point 
in  the  direction  of  B,  the  point  of 
tlie  cu'cle  marked  E,  wiU  have  come  round  to  the  North  end  of  the 
needle,  (since  the  needle  remains  immovable,')  and  the  reading  will 
therefore  be  "  East,"  as  it  should  be.  The  efiect  on  the  reading 
is  the  same  as  if  the  needle  had  moved  to  the  left  the  same  quantity 
which  the  sights  have  moved  to  the  right,  and  the  left  side  is  there- 
fore properly  marked  "  East,"  and  vice  versa.  So,  too,  if  tlie 
bearing  of  che  line  to  C  be  desired,  half-T\-ay  between  North  and 


140  COMPASS  SIRVKVL\G.  [paktii. 

East,  i.  e.  N.  45'^  E.  ;  when  the  sights  and  the  circle  have 
turned  45  degrees  to  the  right,  the  needle,  really  standing  still, 
has  apparently  arrived  at  the  pomt  half-way  between  N.  and  E., 
i.  e.  N.  45°  E. 

Some  sui'veyors'  compasses  are  marked  the  reverse  of  this,  the 
E  on  the  right  and  the  W  en  the  left.  These  letters  must  then  be 
reversed  in  the  mind  before  the  bearing  is  noted  down. 

(237)  Reading  with  Vernier.  "When  the  needle  does  not  pomt 
precisely  to  one  of  the  division  marks  on  the  circle,  the  fractional 
part  of  the  smallest  space  is  usually  estimated  by  the  eye,  as  has 
been  explained.  But  this  fractional  part  may  be  measured  by  the 
Vernier,  described  in  Art.  (229),  as  follows.  Suppose  the  needle 
to  point  between  N.  31^  E.  and  N.  31^°  E.  Turn  the  tangent 
screw,  which  moves  the  compass-box,  till  the  smaller  di\T.sion  (in 
this  case  31°)  has  come  round  to  the  needle.  The  Vernier  wiU 
then  indicate  through  what  space  the  compass-box  has  moved,  and 
therefore  how  much  must  be  added  to  the  reading  of  the  needle. 
Suppose  it  indicates  10  minutes  of  a  degree.  Then  the  bearing  is 
N.  31°  10'  E.  It  is,  however,  so  difficult  to  move  the  Vernier 
without  disturbing  the  whole  instrument,  that  this  is  seldom  resorted 
to  in  practice.  The  chief  use  of  the  Vernier  is  to  set  the  instru- 
ment for  running  lines  and  making  an  allowance  for  the  variation 
of  the  needle,  as  will  be  explained  in  the  proper  place.     A  Vemier- 

A  A^ernier  arc  is  sometimes  attached  to  one  end  of  the  needle 
and  carried  around  by  it. 

(238)  Practical  Hints.  Mark  every  station,  or  spot,  at  which 
the  compass  is  set.  by  driving  a  stake,  or  digging  up  a  sod,  or  piling 
np  stones,  or  other^vise,  so  that  it  can  be  found  if  any  error,  or  other 
cause,  makes  it  necessary  to  repeat  the  survey. 

Very  often  when  the  hne  of  which  the  bearing  is  required,  is  a 
fence,  &c.,  the  compass  cannot  be  set  upon  it.  In  such  cases,  set 
the  compass  so  that  its  centre  is  a  foot  or  two  from  the  line,  and 
set  the  flag-staif  at  precisely  the  same  distance  from  the  hne  at  the 
other  end  of  it.  The  bearing  of  the  flag-staflf  from  the  compass 
will  be  the  same  as  that  of  the  fence,  the  two  lines  being  parallel 


cuAP.  Ill]  The  Field  Work.  141 

The  distances  should  be  measured  on  the  real  line.  If  more  con- 
venient the  compass  may  be  set  at  some  point  on  the  line  prolong- 
ed, or  at  some  intermediate  point  of  the  hue,  "  in  hne"  between  its 
extremities. 

In  setting  the  compass  level,  it  is  more  important-  to  have  it  level 
crosswavs  of  the  sights  than  in  their  direction  ;  since  if  it  be  not  so, 
on  looking  up  or  down  hill  through  the  upper  part  of  one  sight  and 
the  lower  part  of  the  other,  the  line  of  sight  will  not  be  parallel  to 
the  N  and  S,  or  zero  Hne,  on  the  compass,  and  an  incorrect  bear- 
ing will  therefore  be  obtained. 

The  compass  should  not  be  levelled  by  the  needle,  as  some  books 
recommend,  i.  e.  so  levelled  that  the  ends  of  the  needle  shall  be  at 
equal  distances  below  the  glass.  The  needle  should  be  brought  so 
originally  by  the  maker,  but  if  so  adjusted  in  the  morning,  it  will 
not  be  so  at  noon,  o^ving  to  the  daily  variation  in  the  di}).  If 
then  the  compass  be  levelled  by  it,  the  lines  of  sight  w^iU  generally 
be  more  or  less  oblique,  and  therefore  erroneous.  If  the  needle 
touches  the  glass,  when  the  compass  is  levelled,  balance  it  by  sHd- 
ing  the  coil  of  wire  along  it. 

The  same  end  of  the  compass  should  always  go  ahead.  The 
North  end  is  preferable.  The  South  end  will  then  be  nearest  to 
the  observer.  Attention  to  tliis  and  to  the  caution  in  the  next 
paragraph,  will  prevent  any  confusion  in  the  bearings. 

Always  take  the  readings  from  the  same  end  of  the  needle ; 
from  the  North  end,  if  the  North  end  of  the  compass  goes  ahead  ; 
and  vice  versa.  This  is  necessary,  because  tne  two  ends  will  not 
always  cut  opposite  degrees.  With  this  precaution,  however,  the 
angle  of  two  meeting  lines  can  be  obtained  correctly  from  either 
end,  pro\dded  the  same  one  is  used  in  taking  the  bearings  of  both 
the  lines. 

Guard  against  a  very  frequent  source  Fig.  165 

of  error  with  begmners,  in  reading  from  ,^X\\^^^^^^^^^J-^~^X^^^ 
the  wrong  number  of  the  two  between  ^^-^^i?  '  I  ■^/O'^^ 
which  the  needle  points,  such  as  reading  I 

34°  for  26*^,  m  u  case  like  that  in  the 
figure. 


142  €0^1  PASS  SURVEYIIVW.  [part  hi 

Check  the  vibrations  of  the  needle  bj  gently  raising  it  oflf  the 
pivot  so  as  to  touch  the  glass,  and  letting  it  down  again,  by  the  scre-w 
on  the  under  side  of  the  box. 

The  compass  should  be  smartly  tapped  after  the  needle  haa 
settled,  to  destroy  the  effect  of  any  adhesion  to  the  pivot,  or  fric- 
tion of  dust  upon  it. 

AH  iron,  such  as  the  chain,  &c.,must  be  kept  at  a  distance  from 
the  compass,  or  it  will  attract  the  needle,  and  cause  it  to  deviate 
from  its  proper  direction. 

The  surveyor  is  sometimes  troubled  by  the  needle  refusing  to 
traverse  and  adhering  to  the  glass  of  the  compass,  after  he  has 
briskly  wiped  this  off  with  a  silk  handkerchief,  or  it  has  been  car- 
ried so  as  to  rub  against  his  clothes.  The  cause  is  the  electricity 
excited  by  the  friction.  It  is  at  once  discharged  by  applying  a 
wet  finger  to  the  glass. 

A  compass  should  be  carried  with  its  face  resting  against  the 
side  of  the  surveyor,  and  one  of  the  sights  hooked  over  his  arm. 

In  distant  surveys  an  extra  centre  pin  should  be  carried,  (as  it 
is  very  liable  to  injury,  and  its  perfection  is  most  essential),  and, 
also,  an  extra  needle.  When  two  such  are  carried,  they  should 
be  placed  so  that  the  north  pole  of  one  rests  against  the  south  pole 
of  the  other. 

(239)  Wlien  the  magnetism  of  the  needle  is  lessened  or  destroy- 
ed by  time,  it  may  be  renewed  as  follows.  Obtain  two  bar  mag- 
nets. Provide  a  board  with  a  hole  to  admit  of  the  axis,  so  that  its 
collar  may  fit  fairly,  and  that  the  needle  may  rest  flat  on  it,  with- 
out bearing  at  the  centre.  Place  the  board  before  you,  with  the 
north  end  of  the  needle  to  your  right.  Take  a  magnet  in  each 
hand,  the  left  holding  the  North  end  of  the  bar,  or  that  which  haa 
the  mark  across,  downwards  ;  and  the  right  holding  the  same  mark 
upwards.  Brmg  the  bars  over  the  axis,  about  a  foot  above  it, 
without  approaching  each  other  within  two  inches : — bring  them 
down  vertically  on  the  needle,  (the  marks  as  directed)  about 
an  inch  on  each  side  of  its  axis  ;  slide  them  outwards  to  its  ends 
with  slight  pressure ;  raise  them  up  ;  bring  them  to  their  formei 
position,  and  repeat  this  a  number  of  times. 


CHAP.  III.]  The  Field  Work.  143 

(240)  Back  Sights.  To  test  the  accuracy  of  the  bearing  of  a 
line,  taken  at  one  end  of  it,  set  up  the  compass  at  the  other  end, 
or  point  sighted  to,  and  look  back  to  a  rod  held  at  the  first  station, 
or  point  where  the  compass  had  been  placed  originally.  The  read- 
ing of  the  needle  should  now  be  the  same  as  before. 

If  the  position  of  the  sights  had  been  reversed,  the  reading 
would  be  the  Reverse  Bearing ;  a  former  bearing  of  N.  30^  E. 
would  then  be  S.  30°  W.,  and  so  on. 

(241)  Local  attraction.  If  the  Back-sight  does  not  agree 
with  the  first  or  forward  sight,  this  latter  must  be  taken  over  again. 
If  the  same  difference  is  again  found,  this  shows  that  there  is  local 
att-actionni  one  of  the  stations;  i.  e.  some  influence,  such  as  a 
mass  of  iron  ore,  ferruginous  rocks,  &c.,  under  the  surface,  which 
attracts  the  needle,  and  makes  it  deviate  from  its  usual  direction. 
Any  high  object,  such  as  a  house,  a  tree,  &c.,  has  recently  been 
found  to  produce  a  similar  effect. 

To  discover  at  which  station  the  attraction  exists,  set  the  com- 
pass at  several  intermediate  points  in  the  line  which  joins  the  two 
stations,  and  at  points  in  the  line  prolonged,  and  take  the  bearing 
of  the  line  at  each  of  these  points.  The  agreement  of  several  of 
these  bearmgs,  taken  at  distant  points,  will  prove  their  correctness. 
Otherwise,  set  the  compass  at  a  third  station ;  sight  to  each  of  the 
two  doubtful  ones,  and  then  from  them  back  to  this  third  station. 
This  will  show  which  is  correct. 

"Wlien  the  difference  occurs  in  a  series  of  lines,  such  as  around  a 
field,  or  along  a  road,  proceed  Fig-  i^ij. 

thus.    Let  C  be  the  station  at  C_  J^ 

which  the  back-sight  to  B  dif-  v- 
fers  from  the  foresight  from 
B  to  C.  Since  the  back-sight  from  B  to  A  is  supposed  lo  have 
agreed  with  the  foresight  from  A  to  B,  the  local  attraction  must  be 
at  C,  and  the  forward  bearing  must  be  corrected  by  the  difference 
just  found  between  the  fore  and  back  sights,  adding  or  subtracting 
it,  according  to  circumstances.     An  easy  method  is  to  draw  a 


144  COMPASS  SURVKYIIVG.  [part  ui. 

figure  for  the  case,  as  in  Fig.  167.  In 
it,  suppose  the  true  bearing  of  BC,  as 
given  by  a  fore-sight  from  B  to  C,  to  be 
N.  40^  E.,  but  that  there  is  local  at- 
traction at  C,  so  that  the  needle  is  drawn 
aside  10°,  and  points  in  the  direction 
S'N',  instead  of  SN.  The  back-sight 
from  C  to  B  will  then  give  a  bearing 
of  N.  50°  E. ;  a  difference,  or  correc-  <> 
tion  for  the  next  fore-sight,  of  10°.  If  the  next  fore-sight,  from  C 
to  D,  be  N.  70°  E,  this  10°  must  be  subtracted  from  it,  makmg 
the  true  fore-sight  N.  60°  E. 

A  general  rule  may  also  be  given.  When  the  back-sight  is 
greater  than  the  fore-sight,  SiS  in  this  case,  subtract  the  difference 
from  the  next  fore-sight,  if  that  course  and  the  preceding  one  have 
both  their  letters  the  same  (as  in  this  case,  both  being  N.  and  E.), 
or  both  their  letters  different ;  or  add  the  difference  if  either  the 
first  or  last  letters  of  the  two  courses  are  different.  When  the 
hacJc-sight  is  less  than  the  fore-sight,  add  the  difference  in  the  case 
in  which  it  has  just  been  directed  to  subtract  it,  and  subtract  it 
where  it  was  before  directed  to  add  it. 

(242)  Angles  of  deflection.  When  the  compass  indicates 
much  local  attraction,  the  difference  between  the  directions  of 
two  meetmg  lines,  (or  the  "  angle  of  deflection"  of  one  from  the 
other),  can  still  be  correctly  measured,  by  taking  the  difference  of 
the  bearings  of  the  two  lines,  as  observed  at  the  same  point.  For, 
the  error  caused  by  the  local  attraction,  whatever  it  may  be,  affects 
both  bearings  equally,  inasmuch  as  a  "Bearing"  is  the  angle 
which  a  hne  makes  with  the  direction  of  the  needle,  and  that  here 
remains  fixed  in  some  one  direction,  no  matter  what,  during  the 
taking  of  the  two  bearings.  Thus,  in  Fig.  167,  let  the  true  bear- 
ing of  BC,  i.  e.  the  angle  which  it  makes  with  the  Hne  SN,  be,  as 
.  before,  N.  40°  E.,  and  that  of  CD  N.  60°  E.  The  true  "  angle 
of  deflection"  of  these  lines,  or  the  angle  B'CD,is  therefore  20°. 
Now,  if  local  attraction  at  C  causes  the  needle  to  point  in  the  direc- 
S'N',  10°  to  the  left  of  its  proper  direction,  BC  will  bear  N.  50^ 


CHAP.  III.] 


The  Field  Work. 


145 


E.,  and  CD  N.  70^  E.,  and  the  difference  of  these  bearings,  i.  e. 
the  angle  of  deflection,  will  be  the  same  as  before. 

(243)  Angles  between  Courses.  To  determine  the  angle  of 
deflection  of  two  courses  meeting  at  any  point,  the  follo^ving  simple 
rules,  the  reasons  of  which  will  appear  from  the  accompanying 
figures,  are  sufficient. 


Case  1.  When  the  first  letters  of  the 
bearing  are  alike,  (i.  e.  both  N.  or  both 
S.),  and  the  last  letters  also  ahke,  (i.  e. 
both  E.  or  both  W.),  take  the  difference 
of  the  bearmgs.  Example.  If  AB  bears 
N.  30°  E.  and  BC  bears  N.  10°  E.,  the 
ande  of  deflection  CBB'  is  20°. 


Case  2.  When  the  first  letters  are 
alike  and  the  last  letters  different ;  take 
the  sum  of  the  bearings.  Ux.  If  AB 
bears  N.  40°  E.  and  BC  bears  N.  20° 
W. ;  the  angle  CBB'  is  60°. 


Fig.  168. 


vr- 


ritr- 


Fig.  170. 


Case  3.  When  the  first  letters  are 
different  and  the  last  letters  alike,  sub- 
tract the  sum  of  the  bearings  from  180°. 
£x.  If  AB  bears  N.  30°  E.  and  BC 
bears  S.  40°  E. ;  the  angle  CBB' is  110°. 


10 


146 


COMPASS  SURVEYING. 


[part  in 


170. 


Case  4.  When  both  the  first  and 
last  letters  are  different,  subtract  the 
difference  of  the  bearings  from  180°. 
Ex.  If  AB  bears  S.  30°  W.  and  BC 
bears  N.  70°  E. ;  the  angle  CBB'  is 
140° 


If  the  angles  included  between  the  courses  are  desired, 
they  will  be  at  once  found  by  reversing  one  bearing,  and  then  ap- 
plying the  above  rules ;  or  by  subtracting  the  results  obtained  as 
above  from  180° ;  or  an  analogous  set  of  rules  could  be  formed 
for  them. 


(244)  To  change  Bearings.  It  is  convenient  in  certain  cal- 
culations to  suppose  one  of  the  Unes  of  a  survey  to  change  its  direc- 
tion so  as  to  become  due  North  and  South ;  that  is,  to  become  a 
new  Meridian  hne.  It  is  then  necessary  to  determine  what  the 
bearings  of  the  other  lines  will  be,  supposing  them  to  change  with 
it.  The  subject  may  be  made  plain  by  supposing  the  survey  to  be 
platted  in  the  usual  way,  with  the  North  uppermost,  and  the  plat 
to  be  then  turned  around,  till  the  line  to  be  changed  is  in  the  de- 
Bired  direction.  The  effect  of  this  on  the  other  lines  will  be  readily 
seen.     A  G-eneral  Rule  can  also  be  formed. 

Take  the  difference  between  the  original  bearing  of  the  side 
which  becomes  a  Meridian  and  each  of  those  bearings  which  have 
both  their  letters  the  same  as  it,  or  both  different  from  it.  The 
changed  bearings  of  these  lines  retain  the  same  letters  as  before,  if 
they  were  originally  greater  than  the  original  bearing  of  the  new  Me* 
ridian  line  ;  but,  if  they  were  less,  they  are  thrown  on  the  other  side 
of  the  N.  and  S.  hne,  and  their  last  letters  are  changed ;  E.  being 
put  for  W.  and  W  for  E. 

Take  the  sum  of  the  original  bearing  of  the  new  Meridian  line, 
and  each  of  those  bearings  which  have  one  letter  the  same  as  one 
letter  of  the  former  bearing,  and  one  different.    If  this  sum  exceeda 


CHAP     III.] 


The  Field  Hork. 


141 


90^,  this  shews  Jiat  the  line  is  thrown  on  the  other  side  of  tha 
East  or  West  point,  and  the  diflference  between  this  sum  and  180^ 
will  be  the  new  bearing  and  the  first  letter  will  be  changed,  N. 
being  put  for  S.  and  S.  for  N. 

Example.  Let  the  Bearings  of  the  sides  of  a  field  be  as  follows : 
N.  32°  E. ;  N.  80°  E. ;  S.  48°  E. ;  S.  18^  W. ;  N.  73  ^^  w. ; 
North.  Suppose  the  first  side  tc  become  du3  North  ;  the  changed 
bearings  will  then  be  as  follows :  North  ;  N.  48^  E. ;  S.  80°  E. ; 
S.  14°  E. ;  S.  741°  W. ;  N.  32°  W. 

To  apply  the  rule  to  the  "  North"  course,  as  above,  it  must  be 
called  N.  0°  W. ;  and  then  bj  the  Rule,  32°  must  be  added  to  it. 

The  true  bearmgs  can  of  course  be  obtained  from  the  changed 
bearings,  by  reversing  the  operation,  taking  the  sum  instead  of  the 
difference,  and  vice  versa. 


(245)  Line  Surveying^,  This  name  may  be  given  to  surveys 
of  lines,  such  as  the  windings  of  a  brook,  the  curves  of  a  road,  &c., 
by  way  of  distinction  from  Farm  Surveying^  in  which  the  lines 
surveyed  enclose  a  space. 

To  survey  a  brook,  or  any  similar  line,  set  the  compass  at,  or 
near,  one  end  of  it,  and  take  the  bearing  of  an  imaginary  or 
visual  line,  running  in  the  general  average  direction  of  the  brook, 

Fig.  172. 


Buch  as  AB  in  the  figure.  Measure  this  line,  taking  ofisets  to  the 
various  bends  of  the  brook,  as  to  the  fence  explained  in  Art.  (115). 
Then  set  the  compass  at  B,  and  take  a  back-sight  to  A,  and  if 
they  agree,  take  a  fore-sight  to  C,  and  proceed  as  before,  notuig 
particularly  the  pomts  where,  the  line  crosses  the  brook. 

To  survey  a  road,  take  the  bearings  and  lengths  of  the  linea 

Fig.  173. 


u^ 


148  '  COI^IPASS  SURVEYING.  [part  m 

which  can  be  most  conveniently  measured  in  the  road,  and  mea- 
sure offsets  on  each  side,  to  the  Outside  of  the  road. 

When  the  Une  of  a  new  road  is  surveyed,  the  bearings  and 
lengths  of  the  various  portions  of  its  intended  centre  line  should  be 
measured,  and  the  distance  which  it  runs  through  each  man's  land 
should  be  noted.  Stones  should  be  set  in  the  ground  at  recorded 
distances  from  each  aiigle  of  the  line,  or  in  each  line  prolonged  a 
known  distance,  so  as  not  to  be  disturbed  in  making  the  road. 

In  surveying  a  wide  river,  one  bank  may  be  surveyed  by  the 
method  just  given,  and  points  on  the  opposite  banks,  as  trees,  &;c., 
may  be  fixed  by  the  method  of  intersections,  founded  on  the  Fourth 
Method  of  determining  the  position  of  a  point ;  and  fully  explained 
m  Part  IV. 

(246)  Checks  by  intersecting  bearings.  At  each  station  at 
which  the  compass  is  set,  take  bearings  to  some  remarkable  object, 
such  as  a  church  steeple,  a  distant  house,  a  high  tree,  &c.  At 
least  three  bearings  should  be  taken  to  each  object  to  make  it  of 
any  use :  since  two  are  necessary  to  determine  it,  (by  our  Fourth 
Method),  and,  till  thus  determined,  it  can  be  no  check.  When 
the  Hne  is  platted,  by  the  methods  to  be  explained  in  the  next 
chapter,  plat  also  the  lines  given  by  these  bearings.  If  those  taken 
to  the  same  object  from  three  diflferent  stations,  intersect  in  the 
same  point,  this  proves  that  there  has  been  no  mistake  in  the  sur- 
vey or  platting  of  those  stations. 

If  any  bearing  does  not  intersect  a  point  fixed  by  pre\dous  bear- 
ings, it  shows  that  there  has  been  an  error,  either  between  the  last 
station  and  one  of  those  which  fixed  the  point,  or  in  the  last  bear- 
ing to  the  point.  To  discover  which  it  was,  plat  the  following  lint 
of  the  survey,  and,  at  its  extremity,  set  off  the  beaiing  from  it  to  the 
point;  and  if  the  line  thus  platted  passes  through  the  pomt,  it 
proves  that  there  was  no  error  in  the  line,  but  only  in  the  bearing 
to  the  point.  If  otherwise,  the  error  was  somewhere  in  the  line 
between  the  stations  from  which  the  bearings  to  that  point  were 
taken. 


CHAP.  iii.J  The  Field  Work.  149 

(247)  Keeping  (lie  Field-notes.  The  simi.lest  and  easiest 
method  for  a  beginner  is  to  make  a  rough  sketch  of  the  survey  bj 
eje,  and  write  down  on  the  Imes  thtir  bearings  and  lengths. 

An  improvement  on  this  is  to  actually  lay  down  the  precise  bear 
Bigs  and  lengths  of  the  lines  in  the  field-book  in  the  manner  to  be 
explained  in  the  chapter  on  Plattmg,  Art.  (269). 

(248)  A  second  method  is  to  draw  a  straight  line  up  the  page 
of  the  field-book,  and  to  write  on  it  the  bearings  and  lengths  of 
the  lines.  The  only  advantage  of  this  method  is  that  the  line  will 
not  rim  off  the  side  of  the  page,  as  it  is  apt  to  do  in  the  preceding 
method. 

(249)  A  third  method  is  to  represcLi  the  line  surveyed,  by  a 
double  column,  as  in  Part  II,  Chapter  I,  Art.  (95),  wliich  shoidd 
be  now  referred  to.  The  bearings  are  written  obliquely  up  the 
columns.  At  the  end  of  each  course,  its  length  is  written  in  the 
column,  and  a  line  drawn  across  it.  Dotted  Unes  are  drawn  across 
the  column  at  any  intermediate  measurement.  Offsets  are  noted 
as  explained  in  Art.  (114). 

The  intersection-bearings,  described  in  Art.  (246),  should  be 
entered  in  the  field-book  before  the  bearings  of  the  line,  in  order 
to  avoid  mistakes  of  platting,  in  setting  off  tlie  measured  distances 
on  the  wrong  line. 

(250)  A  fourth  method  is  to  write  the  Stations,  Beaiings,  and 
Distances  in  three  columns.  This  is  compact,  and  has  the  advan- 
tage, when  apphed  to  farm  surveying,  of  presenting  a  form  suitable 
for  the  subsequent  calculations  of  Content,  but  does  not  give  facili- 
ties for  noting  offsets. 

Examples  of  these  four  methods  are  given  m  Art.  (254)  ;  which 
contains  the  field-notes  of  the  lines  bounding  a  field.  i 

(251)  IVew-York  Canal  Maps.  The  following  is  a  description 
of  the  original  maps  of  the  survey  of  the  line  of  the  New- York  Erie 
Canal,  as  published  by  the  Canal  Commissioners.  The  figure 
represents  a  portion  of  such  a  map ;  but,  necessarily,  with  all  its 
Unes  black  ;  r&l  and  blue  Imes  being  used  on  the  real  map. 


150 


COMPASS  SURFETING. 


fPART  III 


Fi-.  174. 


s  irEis  zrBls  3°  e]s  34°e 


E   S   5t°E 


"  The  Red  Line  described  along  the  inner  edge  of  the  towing 
path  is  the  base  line,  upon  which  all  the  measurements  in  the  du'ec- 
tion  of  the  length  of  the  canal  were  made.  The  bearings  refer  to 
the  magnetic  meridian  at  the  time  of  the  sm-vey.  The  lengtlis  of 
the  several  portions  are  inserted  at  the  end  of  each,  in  chains  and 
links.  The  offsets  at  each  station  are  represented  bj  red  Jnes 
dra"\Yn  across  the  canal  in  such  a  direction  as  to  bisect  the  angles 
formed  by  the  two  contiguous  portions  of  the  red  or  base  Une,  upon 
the  towing  path.  The  intermediate  oflfsets  are  set  oflF  at  right  angles 
to  the  base  line ;  and  the  distances  on  both  are  given  from  it  in 
links.  The  intermediate  offsets  are  represented  by  red  dotted  lines, 
and  the  distances  to  them  upon  the  base  line  are  reckoned,  m  each 
case,  from  the  last  preceding  station.  The  same  is  likewise  done 
with  the  other  distances  upon  the  base  line  ;  those  to  the  Bridges 
being  taken  to  the  lines  joining  the  nearest  angles,  or  corner  posts 
of  their  abutments  ;  those  to  the  Lochs  extending  to  the  lines  pass- 
ing thi'ough  the  centres  of  the  two  nearest  quoin  posts ;  and  those 
to  the  Aqueducts,  to  the  faces  of  their  abutments.  The  space 
enclosed  by  the  Blue  Lines  represents  the  portion  embraced  with- 
in the  limits  of  the  survey  as  belonging  to  the  state  ;  and  the  names 
of  the  adjoining  proprietors  are  given  as  they  stood  at  the  time  of 
executing  the  survey.  The  distances  are  projected  upon  a  scale 
of  two  chains  to  the  inch." 


(252)  Farm  Surveying.  A  farm,  or  field,  or  other  space  m- 
cluded  within  known  lines,  is  usually  surveyed  by  the  compass 
thus.  Begin  by  walking  around  the  boundary  lines,  and  setting 
stakes  at  all  the.  corners,  which  the  flag-man  should  specially  note, 


CHAP.  Ill]  The  Field  Work,  151 

80  that  he  may  readily  find  them  again.  Then  set  the  compass  at 
any  corner,  and  send  the  flag-man  to  the  next  corner.  Take  the 
bearing  of  the  bounding  line  running  from  corner  to  corner,  which 
is  usually  a  fence.  Measure  its  length,  taking  offsets  if  necessary. 
Note  where  any  other  fence,  or  road,  or  other  line,  crosses  or  meets 
it,  and  take  their  bearings.  Take  the  compass  to  the  end  of  this 
first  bounding  line  ;  sight  back,  and  if  the  back-sight  agrees,  take 
the  bearmg  and  distance  of  the  next  bounding  Hne  ;  and  so  proceed 
till  you  have  got  back  to  the  point  of  starting. 

(253)  Where  speed  is  more  important  than  accuracy  hi  a  sur- 
vey, whether  of  a  line  or  a  farm,  the  compass  need  be  set  only  at 
every  other  station,  taking  a  forward  sight;  from  ihe  1st  station  to 
the  2d  ;  then  settmg  the  compass  at  the  3d  station,  takmg  a  back' 
Bight  to  the  2d  station  (but  with  the  north  point  of  the  compass  al- 
ways ahead),  and  a  fore-sight  to  the  4th ;  then  gouig  to  the  5th, 
and  so  on.     This  is,  however,  not  to  be  recommended. 

(254)  Field-notes,  The  Field-notes  of  a  Farm  survey  may  be 
kept  by  any  of  the  methods  which  have  been  described  with  lefer- 
ence  io  a  Line  survey.  Below  are  given  the  Field-notes  of  the 
game  field  recorded  by  each  of  the  methods. 

First  Method. 

Fig.  175. 


152 


COMPASS  SURVEYING. 


[part  m 


Second 

TJdrd 

Method. 

Method. 

0(1) 

-(1)- 

^ 

3.23 

o 

CD 

to 

CO 
CO 

^ 

to 

0(5) 

^ 

^ 

-(5)- 

o 

LO 

3.54 

-hM- 

to 

CO 

CO 

^ 

-hw 

CQ 

S!     1 

( 

)(4) 

CO        1 

m 

p4 

-(4)- 

o 

G<l 

2.22 

lO 

c4 

F^ 

QQ 

t- 

C 

)(3) 

tO 

m 

p4 

-(3)- 

o 

CO 

05 

1.29 

00 
J25 

tH 

CO 

0(2) 

GO 

m 

^ 

o 

O 

-(2)- 

CO 

2.70 

52^ 

m 

0(1) 

to 

CO 

^ 

-(1)- 

Fourth  Method. 


'sTATIOIfS.I   BEARINGS. 

DISTANCES. 

1 

2 
3 
4 
5 

N.  350    E. 
N.  8310  E. 
S.  570    E. 
S.  34|°W. 
N.561CW. 

2.70 

1.29 
2.22 
3.55 
3.S3 

(^255)  The  Field-notes  of  a  field,  in  which  offsets  occur,  may  bie 
most  easily  recordedby  the  Third  Method ;  as  ui  Fig.  176. 

Wlien  the  Field-notes  are  recorded  by  the  Fourth  Method, 
the  offsets  may  be  kept  in  a  separate  Table;  in  which  the  1st 
eoumn  will  contain  the  stations  from  which  the  measurements  are 
made,  the  2d  column  the  distances  at  which  they  occur,  the  3d 

*  In  the  "  Third  Method,"  the  bearings  should  be  written  obliquely  upward 
ks  directed  in  Art,  (249)  but  artf  not  so  printed  here,  from  typographical  diffi 
calties, 


CHAP  III. J  The  Field  WorK.  J  52 

column  the  lengths  of  the  offsets,  and  the  4th  column  the  side  of 
the  line,  "  Right,"  or  "  Left,"  on  which  they  lie. 

For  calculation,  four  more  columns  may  be  added  to  the  table, 
containing  the  intervals  between  the  oifsets  ;  the  sums  of  the 
adjoining  pairs  ;  and  the  products  of  the  numbers  in  the  two  pro- 
ceding  colunms,  separated  into  Right  and  Left,  one  being  additiv3 
to  the  field,  and  the  other  subtractive. 

(256)  Tests  of  accuracy.  1st.  The  check  of  intersections  d©- 
<5cribed  in  Art.  (246),  may  be  employed  to  great  advantage,  when 
some  conspicuous  object  near  the  centre  of  the  farm  can  be  seen 
from  most  of  its  corners. 

2nd.  When  the  survey  is  platted,  if  the  last  course  meets  the 
starting  point,  it  proves  the  work,  and  the  survey  is  then  said  to 
"  close." 

3d.  Diagonal  lines,  running  from  corner  to  corner  of  the  farm, 
like  the  "  Proof-lines"  in  Chain  Surveying,  may  be  measured  and 
their  bearings  taken.  When  these  are  laid  down  on  the  plat,  their 
meeting  the  points  to  which  they  had  been  measured,  proves  the 
work. 

4th.  The  only  certain  and  precise  test  is,  however,  that  by 
"  Latitudes  and  Departures."  This  is  fully  explahaed  in  Chapter 
V,  of  this  Part. 

(2.57)  A  very  fallacious  test  is  recommended  by  several  writers 
en  this  subject.  It  is  a  well-known  proposition  of  Geometry,  that 
in  any  figure  bounded  by  straight  lines,  the  sum  of  all  the  interior 
angles  is  equal  to  twice  as  many  right  angles,  as  the  figure  has  sides 
less  two ;  since  the  figure  can  be  divided  into  that  number  of  tri- 
angles. Hence  this  common  rule.  "  Calculate  [by  the  last  para- 
graph of  Art.  (243)]  the  interior  angles  of  the  field  or  farm  sur- 
veyed ;  add  them  together,  and  if  their  sum  equals  twice  as  many 
right  angles  as  the  figure  has  sides  less  two,  the  angles  have  been 
correctly  measured."  This  rule  is  not  applicable  to  a  compass  sur- 
vey ;  for,  in  Fig.  167,  page  144,  the  interior  angle  BCD  vdW  con- 
tain tlie  same  number  of  degrees  (m  that  case  160^)  whether  the 
bearings    of   the  sides  have  been  noted  correctly,  as  being  the 


154  CO:HPASS  SIRVE¥I\G.  [pari  in 

argles  which  they  make  with  NS — or  mcorrectly,  as  being  the 
angles  winch  they  make  with  N'S'.  This  rule  would  therefore 
prove  the  work  in  either  case. 

(258)  Method  of  Radiation.  A  fidd  may  he  surveyed  from 
one  station,  either  within  it  or  without  it,  by  taking  the  bearings  and 
the  distances  from  that  point  to  each  of  the  comers  of  the  field. 
These  corners  are  then  "  determined,"  by  the  3d  method,  Art.  (7). 
This  modification  of  that  method,  we  named,  in  Art.  (220),  the 
Method  of  Radiation.  All  our  preceding  surveys  with  the  com- 
pass have  been  by  the  Method  of  Progression. 

The  compass  may  be  set  at  one  corner  of  the  field,  or  at  a  point 
in  one  of  its  sides,  and  the  same  method  of  Radiation  employed. 

This  method  is  seldom  used  however,  since,  unlike  the  method 
of  Progi-ession,  its  operations  are  not  checks  upon  each  other. 

(259)  Metliod  of  Intersection.  A  field  may  also  be  surveyed 
by  measuring  a  base  line,  either  within  it  or  without  it,  setting  the 
compass  at  each  end  of  the  base  Hue,  and  taking,  from  each  end, 
the  bearings  of  each  corner  of  the  field ;  which  will  then  be  fixed 
and  determined,  by  the  4th  method.  Art.  (8).  This  mode  of  sur 
veying  is  the  Method  of  Intersections,  noticed  in  Art.  (220).  It 
win  be  fully  treated  of  m  Part  V,  under  the  title  of  Triangular 
Surveying 

(260)  Running  out  old  lines.  The  original  surveys  of  lands 
in  the  older  States  of  the  American  Union,  were  exceedingly  defi- 
cient in  precision.  This  arose  from  two  principal  causes  ;  the  small 
value  of  land  at  the  period  of  these  surveys,  and  the  want  of  skill 
in  the  surveyors.  The  effect  at  the  present  day  is  frequent  dissat- 
isfaction and  htigation.  Lots  sometimes  contain  more  acres  than 
they  were  sold  for,  and  sometimes  less.  Lines  which  are  straight 
in  the  deed,  and  on  the  map,  are  found  to  be  crooked  on  the 
ground.  The  recorded  surveys  of  two  adjoming  farms  often  make 
one  overlap  the  other,  or  leave  a  gore  between  them.  The  most 
difficult  and  delicate  duty  of  the  land-surveyor,  is  to  run  out  these 
old  boundary  lines.     In  such  cases,  his  first  business  is  to  find 


CHA^.   III.] 


The  Field  Work. 


155 


monuments,  stones,  marked  trees,  stumps,  or  any  other  old  "  cor- 
ners," or  landmarks.  These  are  his  starting  points.  The  o-wnerg 
whose  lands  join  at  these  corners  should  agree  on  them.  Old 
fences  must  generally  be  accepted  by  right  of  possession ;  though 
such  questions  belong  rather  to  the  lawyer  than  to  the  surveyor.* 
His  business  is  to  mark  out  on  the  ground  the  hnes  given  in  the 
deed.  When  the  bounds  are  given  by  compass-bearings,  the  sur- 
veyor must  be  remuided  that  these  bearings  are  very  far  from  being 
the  same  now  as  originally,  having  been  changing  every  year. 
The  method  of  deternuning  this  important  change,  and  of  making 
the  proper  allowance,  will  be  found  in  Chapter  VIII,  of  this  Part. 

(261)  Town  Surveying".  Begin  at  the  meeting  of  two  or  more 
of  the  principal  streets,  through  which  you  can  have  the  longest 
prospects.  Having  fixed  the  instrument  at  that  point,  and  taken 
the  bearings  of  all  the  streets  issuing  from  it,  measure  all  these  lines 
with  the  chain,  taking  offsets  to  all  the  corners  of  streets,  lanes, 
benduigs,  or  windings  ;  and  to  all  remarkable  objects,  as  churches, 
markets,  pubUc  buildings,  &c.  Then  remove  the  instrument  to 
the  next  street,  take  its  bearings,  and  measure  along  the  street  aa 
before,  taking  offsets  as  you  go  along,  with  the  offset-staff.  Proceed 
in  this  manner  from  street  to  street,  measuring  the  distances  and 
offsets  as  you  proceed. 


"  "  In  the  description  of  laud  conveyed,  the  rule  i^,  that  known  and  fixed  mon* 
Qinents  control  courses  and  distances.  So,  the  certainty  of  metes  and  bounds  will 
include  and  pass  all  the  lands  within  them,  though  they  vary  (rom  the  ^ven 
quantity  expressed  in  the  deed.  In  New-York,  to  remove,  deface  or  alter  land 
marks  maliciously    is  an  indictable  offence." — Kent^s  Commentaries,  IV,  515 


156  €OilIPASS  SURVEYING.     .  [partiu 

Thus,  ill  the  figure,  fix  the  instrument  at  A,  and  measure  linea 
in  the  direction  of  all  the  streets  meeting  there,  noting  their  bear- 
ings ;  then  measure  AB,  noting  the  streets  at  X,  X.  At  the  second 
station,  B,  take  the  bearhigs  of  all  the  streets  which  meet  there  ; 
and  measure  from  B  to  C,  noting  the  places  and  the  bearings  of 
all  the  cross-streets  as  you  pass  them.  Proceed  m  hke  manner 
from  C  to  D,  and  from  D  to  A,  "  closing"  there,  as  in  a  farm  sur- 
vey. Having  thus  surveyed  all  the  principal  streets  in  a  particu- 
lar neighborhood,  proceed  then  to  survey  the  smaller  intermediate 
streets,  and  last  of  all,  the  lanes,  alleys,  courts,  yards,  and  every 
other  place  wliich  it  may  be  thought  proper  to  represent  in  the 
plan.  The  several  cross-streets  answer  as  good  check  lines,  to 
prove  the  accuracy  of  the  work.  In  this  manner  you  continue  till 
you  take  in  all  the  town  or  city. 

(262)  Obstacles  in  Compass  Surveying.  The  various  obsta- 
cles which  may  be  met  with  in  Compass  Surveying,  such  as  woods, 
water,  houses,  &c.,  can  be  overcome  much  more  easily  than  in 
Chain  Surveying.  But  as  some  of  the  best  methods  for  effecting 
this  involve  principles  which  have  not  yet  been  fully  developed,  it 
will  be  better  to  postpone  giving  any  of  them,  till  they  can  be  all 
treated  of  together ;  which  will  be  done  in  Part  VII. 


CHAPTER  IV. 


PLATTING  THE  SURVEY. 

(263)  The  platting  of  a  survey  made  with  the  3ompas3,  consiste 
in  drawing  on  paper  the  lines  and  the  angles  which  have  been 
measured  on  the  ground.  The  luies  are  drawn  "  to  scale,"  as  haa 
been  fuUj  explained  in  Part  I,  Chapter  III.  The  manner  of  plat- 
ting angles  was  referred  to  in  Art.  (41),  but  its  explanation  has 
been  reserved  for  this  place. 


(264)  With  a  Protractor.  A  Protractor  is  an  mstmmem 
made  for  this  object,  and  is  usually  a  semicircle  of  brass,  as  in  the 
figure,  with  its  semi-circumference  divided  into  180  equal  parts,  or 

Fig.  178. 


degrees,  and  numbered  in  both  directions.  It  is,  in  fact,  a  aoinia^ 
ture  of  the  instrument,  (or  of  half  of  it),  -with  which  the  angles 
have  been  measured.  To  lay  oflf  any  angle  at  any  point  cf  a 
straight  line,  place  the  Protractor  so  that  its  straight  side,"'  the 
diameter  of  the  semi-circle,  is  on  the  given  line,  and  the  middle  of 
this  diameter,  which  is  marked  by  a  notch,  is  at  the  given  point. 
With  a  needle,  or  sharp  pencil,  make  a  mark  on  the  paper  at  the 
reqmred  number  of  degrees,  and  draw  a  line  from  the  mark  to  the 
given  point. 


158 


COMPASS  SURVEYING. 


[part  III 


Sometimes  the  protractor  has  an  arm  turning  on  its  centre,  and 
extending  bejond  its  circumference,  so  that  a  line  can  be  at  once 
drawn  by  it  when  it  is  set  to  the  desired  angle.  A  Vernier  scale 
is  sometimes  added  to  it  to  increase  its  precision. 

A  Rectangular  Protractor  is  sometimes  used,  the  divisions  of 
degrees  being  engraved  along  three  edges  of  a  plane  scale.  The 
semi-circular  one  is  preferable.  The  objection  to  the  rectangular 
protractor  is  that  the  division  correspondmg  to  a  degree  is  very 

Fig.  179. 


V     \\\\\\\\|IM///////     ^~y 
JLO    \    &n\gO\?0\80  190tlOOIH)/120/lSO  /    14^0    /      ].&D 

140  180  li^aioidoaloao  7d  eft    i5     io        Z«b 


so     \ 


-a 


^5J)         140    130li^]imd0  9|080  7^  gft     /O 


ySQ 


unequal  on  different  parts  of  the  scale,  being  usually  two  or  three 
times  as  great  at  its  ends  as  at  its  middle. 

A  Protractor  embracing  an  entire  circle,  with  arms  carrying 
verniers,  is  also  sometimes  employed,  for  the  sake  of  greater  accu- 
racy. 


(265)  Platting  Bearing's.  Since  "  Bearings "  taken  -with  the 
Compass  are  the  angles  which  the  various  lines  make  with  the 
^lagnetic  Meridian,  or  the  direction  of  the  compass-needle,  which, 
as  we  have  seen,  remams  always  (approximately)  parallel  to  itself, 
it  is  necessary  to  draw  these  meridians  through  each  station,  before 
laying  off  the  angles  of  the  bearmgs. 

The  T  square,  shown  in  Fig.  14,  is  the  most  convenient  instru- 
ment for  this  purpose.  The  paper  on  which  the  plat  is  to  be  made 
IS  fastened  on  the  board  so  that  the  intended  direction  of  the 
North  and  South  line  may  be  parallel  to  one  of  the  sides  of  the 
board.  The  inner  side  of  the  stock  of  the  T  square  bemg  pressed 
against  one  of  the  other  sides  of  the  board  and  slid  along,  the  edge 
of  the  long  blade  of  the  square  will  always  be  parallel  to  itself  and 
to  the  first  named  side  of  the  board,  and  will  thus  represent  the 
meridian  passing  through  any  station. 


:hap.  IV.] 


Platting  the  Survey. 


159 


If  a  sti-aight-edged  drawing  Fig.  180. 

board  or  table  cannot  be  pro- 
cured, nail  dovra  on  a  table  of 
any  shape  a  straight-edged  ru- 
ler, and  slide  along  against  it 
the  outside  of  the  stock  of  a  T 
square,  one  side  of  the  stock 
being  flush  with  the  blade. 

A  parallel  ruler  may  also  be 
used,  one  part  of  it  being 
screwed  down  to  the  board  in 
the  proper  position. 

If  none  of  these  means  are  at  hand,  approximately  parallel  meri- 
dians may  be  drawn  by  the  edges  of  a  common  ruler,  at  distances 
apart  equal  to  its  width,  and  the  diameter  of  the  protractor  made 
parallel  to  them  by  measuring  equal  distances  between  it  and  them. 


(266)  To  plat  a  survey  with  these  instruments,  mark,  with  a  fine 
point  enclosed  in  a  circle,  a  convenient  spot  in  the  paper  to  repre- 
lent  the  first  station,  1  in  the  figure.     Its  place  must  be  so  chosen 

Fig.  181. 


160  COMPASS  SURVEYING.  Ipart  m 

that  the  plat  may  not  "  run  off"  the  paper.  "With  the  T  square 
draw  a  meridian  through  it.  The  top  of  the  paper  is  usually, 
though  not  necessaiily,  called  North.  With  the  protractor  lay  off 
the  angle  of  the  first  bearing,  as  directed  in  Art.  (264).  Set  off 
the  length  of  the  first  Ime,  to  the  desired  scale,  by  Art.  (42),  from 
1  to  2.     The  line  1 2  represents  the  first  course. 

Through  2,  draw  another  meridian,  lay  off  the  angle  of  the 
second  course,  and  set  off  the  length  of  this  course,  from  2  to  3. 

Proceed  in  like  manner  for  each  course.  When  the  last  course  is 
platted,  it  should  end  precisely  at  the  starting  point,  as  the  survey 
did,  if  it  were  a  closed  survey,  as  of  a  field.  If  the  plat  does  not 
"  close,"  or  "  come  together,"  it  shows  some  error  or  inaccuracy 
either  in  the  original  survey,  if  that  have  not  been  "  tested "  by 
Latitudes  and  Departures,  or  in  the  work  of  platting.  A  method 
of  correction  is  explained  in  Art.  (268).  The  plat  here  given  is 
the  same  as  that  of  Fig.  175,  page  151. 

This  manner  of  laying  down  the  directions  of  lines,  by  the  angles 
which  they  make  with  a  meridian  line,  has  a  great  advantage,  in 
both  accuracy  and  rapidity,  over  the  method  of  platting  lines  by 
the  angles  which  each  makes  with  the  line  which  comes  befcre  it. 
In  the  latter  method,  any  error  in  the  direction  of  one  line  makes 
all  that  follow  it  also  wrong  in  their  directions.  In  the  former,  the 
direction  of  each  line  is  independent  of  the  preceding  line,  though 
its  position  would  be  changed  by  a  previous  error. 

Instead  of  drawing  a  meridian  through  each  station,  sometimes 
only  one  is  dra^vn,  near  the  middle  of  the  sheet,  and  all  the  bear- 
ings of  the  survey  are  laid  off  from  some  one  point  of  it,  as  shown 
in  the  figure,  and  numbered  to  correspond  with  the  stations  from 
which  these  bearings  were  taken.  The  circular  protractor  is  conve- 
nient for  this.  They  are  then  transferred  to  the  places  where 
they  are  wanted,  by  a  triangle  or  other  parallel  ruler,  as  explained 
on  page  27.  The  figure  at  the  top  of  the  next  page  represents 
the  same  field  platted  by  this  method. 

A  semi-circular  protractor  is  sometimes  attached  to  the  stock 
end  of  the  T  square,  so  that  its  blade  may  be  set  at  any  desired 
angle  with  the  meridian,  and  any  bearing  be  thus  protracted  with- 
out drawing  a  meridian.     It  has  some  inconveniences. 


;hap.  rr 


Platting  the  Surrey. 


161 


S     0 


(267)  The  Compass  itself  may  be  used  to  plat  bearings.  For 
this  purpose  it  must  be  attached  to  a  square  board  so  that  the  N 
and  S  Hne  of  the  compass  box  may  be  parallel  to  two  opposite 
edges'of  the  board.  This  is  placed  on  the  paper,  and  the  box  is 
turned  till  the  needle  points  as  it  did  when  the  first  bearing  was 
taken.  Then  a  Une  drawn  by  one  edge  of  the  b'Dard  will  be  in  a 
proper  direction.  Mark  off  its  length,  and  plat  the  next  and  the 
succeeding  bearings  in  the  same  manner. 


-+ 


(268)  When  the  plat  of  a  survey  does  not  "  close,"  it  may  be 
corrected  as  follows.     Let  Fig.  i83. 

ABODE  be  the  boundary 
lines  platted  according  to 
the  given  bearmgs  and 
distances,  and  suppose  that 
the  last  course  comes  to  E, 
instead  of  ending  at  A,  as 
it  should.  Suppose  also 
that  there  is  no  reason  to 
suspect  any  single  great 
error,  and  that  no  one  of  the  lines  was  measured  over  very  rough 

11 


A<<1 


162 


COMPASS  SURVEYIIVG. 


fPART  ;iJ 


ground,  or  was  specially  uncertain  in  its  direction  wlien  observed. 
The  inaccuracy  must  then  be  distributed  among  all  the  lines  in 
■proportion  to  their  length.  Each  point  in  the  figure,  B,  C,  D,  E,  must 
be  moved  in  a  direction  parallel  to  EA,  by  a  certain  distance  which 
is  obtamed  thus.  Multiply  the  distance  EA  by  the  distance  AB, 
and  divide  by  the  sum  of  all  the  courses.  The  quotient  ydYL  be  the 
distance  BB'.  To  get  CC,  multiply  EA  by  AB  +  BC,  and  divide 
the  product  by  the  same  sum  of  aU  the  courses.  To  get  DD',  mul- 
taply  EA  by  AB  +  BC  +  CD,  and  divide  as  before.  So  for  any 
course,  multiply  by  the  sum  of  the  lengths  of  that  course  and  of  all 
those  preceding  it,  and '  divide  as  before.  Join  the  pomts  thus 
obtained,  and  the  closed  polygon  AB'C'D'A  will  thus  be  formed, 
and  wiU  be  the  most  probable  plat  of  the  given  survey.* 

The  method  of  Latitudes  and  Departures,  to  be  explained  here 
after,  is,  however,  the  best  for  effecting  this  object. 

(269)  Field  Platting".  It  is  sometimes  desirable  to  plat  the 
courses  of  a  survey  in  the  field,  as  soon  as  they  are  taken,  as  was 
mentioned  in  Art.  (247),  under  the  head  of  "  Keepuig  the  field- 
notes."  One  method  of  doing  this  is  to  have  the  paper  of  the 
Field-book  ruled  with  parallel  lines,  at  unequal  distances  apart, 
and  to  use  a  rectangular  pro- 
tractor (which  may  be  made 
of  Bristol-board,  or  other  stout 
drawing  paper,)  with  lines  rul- 
ed across  it  at  equal  distances 
of  some  fraction  of  an  inch.  A 
bearing  having  been  taken  and 
noted,  the  protractor  is  laid  on 
the  paper  and  its  centre  placed  at  the  station  where  the  bearing  is 
to  be  laid  off.  It  is  then  turned  tiU  one  of  its  cross-lines  coincides 
with  some  one  of  the  luies  on  the  paper,  which  represent  East  and 
West  fines.  The  long  side  of  the  protractoi  wiU  then  be  on  a 
meridian  and  the  proper  angle  (40°  in  the  figure)  can  be  at  once 
marked  off.  The  length  of  the  course  can  also  be  set  off  by  the 
equal  spaces  between  the  cross-lines,  letting  each  space  represent 
any  convenient  number  of  finks. 

•  This  was  demonstrated  by  Dr.  Bowditch,  in  No.  4,  of  "  The  Analyst.*- 


CHAP.   IV.] 


Platting  the  Survey. 


163 


(270)  A  common  rectangular  protractor  without  any  cross-lines, 
or  a  semi-circular  one,  can  also 
be  used  for  the  same  purpose. 
The  parallel  lines  on  the  paper 
(which,  in  this  method,  may 
be  equi-distant,  as  in  common 
ruled  writing  paper)  will  now 
represent  meridians.  Place 
the  centre  of  the  protractor 
on  the  meridian  nearest  to  the 
station  at  which  the  angle  is  to 
be  laid  off,  and  turn  it  till  the 
given  number  of  degrees  is  cut  by  the  meridian.  Shde  the  pro- 
tractor up  or  down  the  meridian  (which  must  continue  to  pass 
through  the  centre  and  the  proper  degree)  till  its  edge  passes 
through  the  station,  and  then  draw  by  this  edge  a  hne,  which  will 
have  the  bearing  required. 


F 

•g-.. 

18J 

4 

/ 

/ 

0 

if 

^/ 

/ 

1^,^^ 

- 

% 

\// 

? 

/ 

^ 

Hv. 

A 

/ 

/ 

^y/ 

^' 

_. 

(271)  Paper  ruled  into  squares,  (as  are  sometimes  the  right- 
hand  pages  of  surveyors'  field-books),  may  be  used  for  platting 
bearings  in  the  field.  The  lines  running  up  the  page  may  be  called 
North  and  South  lines,  and  those  running  across  the  page  will  then 
be  East  and  West  lines.  Any  course  of  the  survey  will  be  the 
hypothenuse  of  a  right-angled  triangle,  and  the  ratio  of  its  other 
two  sides  will  determine  the 
angle.  Thus,  if  the  ratio  of 
the  two  sides  of  the  right-an- 
gled triangle,  of  which  the  line 
AB  in  the  figure  is  the  hypoth- 
enuse, is  1,  that  line  makes  an 
angle  of  45°  with  the  meridian. 
If  the  ratio  of  the  long  to  the 
short  side  of  the  right-angled 
triangle  of  which  the  line  AC 
is  the  hypothenuse,  is  4  to  1, 
the  line  AC  makes  an  angle 
of  14°  with  the  meridian.     The  line  AD,  the  hypothenuse  of  an 


Fig 

186. 

c 

B 

/ 

i 

1 

/ 

/ 

/ 

// 

E 

/ 

u 

/ 

/ 

1 

• 

/ 

// 

/ 

-D- 

1/ 

/ 

^ 

-^ 

"^ 

^ 

|/ 

-^ 

^ 

— 

A 

1 
1 

164 


COMPASS  SURVEYIIVG. 


[part  III 


equal  triangle,  which  has  its  long  side  lying  East  and  West,  makes 
likewise  an  ande  of  14"  with  that  side,  and  therefore  makes  an 
angle  of  76°  with  the  meridian.* 

To  facilitate  the  use  of  this  method,  the  following  table  has  b^>en 
prepared. 

TABLE  FOR  PLATTING  BY  SQUARES. 


1-? 

Ratio  of 

'J 

°r 

lone;  side  to 

0  "^ 

S  o 

c  '^ 
< 

short  fide. 

Tr!_3 

< 

1° 

57  3  to  1 

89- 

2° 

28.6  to  1 

88° 

3° 

19.1  to  1 

87° 

40 

14.3  to  1 

86° 

5^ 

11.4  to  1 

85° 

6° 

9.5  to  1 

84° 

70 

8.1  to  1 

83° 

8° 

7.1  to  1 

82° 

90 

6.3  to  1 

81° 

10° 

5.7  to  1 

80° 

11° 

5.1  to  1 

79- 

12° 

4.9  to  1 

78° 

13° 

4.3  to  1 

77° 

14° 

4.0  to  1 

76° 

15° 

3.7  to  1 

75= 

0  ^ 

S  5 

Ratio  of 
long  side  to 
sliort  side. 

d  5; 

a— 

< 

16- 

3.49  to  1 

74- 

17° 

3.27  to  1 

73° 

18° 

3.08  to  1 

72° 

19° 

2.90  to  1 

71° 

20° 

2.75  to  1 

70° 

21° 

2.61  to  1 

69° 

22° 

2.48  to  1 

68° 

23° 

2.36  to  1 

67° 

24° 

2.25  to  1 

66° 

25° 

2.14  to  1 

65° 

26° 

2.05  to  1 

64° 

27° 

1.96  to  1 

63° 

28° 

1.88  to  1 

62° 

29° 

1.80  to  1 

61° 

30O 

1.73  to  1 

60° 

0 

g  a; 

n't: 
< 

Ratio  of 
lonar  side  to 
short  side. 

B    1 

ac    J,- 

£.2 

C  *i 

S  3 
< 

31° 

1.664  to  1 

59° 

32° 

1.600  to  1 

58° 

33° 

1.540  to  1 

57° 

34° 

1.483  to  1 

56° 

35° 

1.428  to  1 

55° 

36^ 

1.376  to  1 

54° 

37° 

1.327  to  1 

53° 

38° 

1.280  to  1 

52° 

39° 

1.235  to  1 

51° 

40° 

1.192  to  1 

50° 

41° 

1.150  to  1 

49° 

42° 

1.111  to  1 

48° 

43° 

1.072  to  1 

47° 

44° 

1.036  to  1 

46° 

45° 

1.000  to  1 

45° 

To  use 'this  table,  find  in  it  the  ratio  corresponding  to  the  angie 
which  you  wish  to  plat.  Then  count,  on  the  ruled  paper,  any 
number  of  squares  to  the  right  or  to  the  left  of  the  point  which 
represents  the  station,  according  as  your  bearing  was  East  or  West ; 
and  count  upward  or  downward  according  as  your  bearing  was  North 
or  South,  the  number  of  squares  given  by  multiplying  the  first  num 
ber  by  the  ratio  of  the  Table.  Thus  ;  if  the  given  bearing  from  A 
in  the  figure,  was  N.  20°  E.  and  two  squares  were  counted  to  the 
right,  then  2  x  2.75  =  5|  squares,  should  be  counted  upward,  to 
E,  and  AE  would  be  the  required  course. 


(272)  With  a  paper  protractori  Engraved  paper  protractors 
may  be  obtained  from  the  instrument-makers,  and  are  very  conve^ 

*  This  and  all  the  following  ratios  may  be  obtained  directly  from  Trigonome. 
ancal  Tables  ;  for  the  "ratio  of  the  long  "side  to  the  short  side,  the  latter  beiii$ 
iaken  a«  unity,  is  the  natural  cotangent  of  the  angle. 


IV.] 


Platting  the  SurTey. 


1()5 


nient.  A  circle  of  large  size,  divided  into  degrees  and  quartei-a, 
is  engraved  on  copi^er,  and  impressions  from  it  are  taken  on  draw- 
ing paper.  The  divisions  are  not  numbered.  Draw  a  straight  line 
to  represent  a  meridian,  chrough  the  centre  of  the  circle,  in  any 
convenient  direction.  Number  the  degrees  from  0  to  90°,  each 
way  from  the  ends  of  this  meridian,  as  on  the  compass-plate.  The 
protractor  is  now  ready  for 
use.  Choose  a  convenient 
point  for  the  first  station. 
Suppose  the  first  bearing  to 
be  N.  30°  E.  The  line  pass- 
ing; through  the  centre  of  the 
circle  and  through  the  oppo- 
site points  N.  30°  E.  and  S. 
30°  W.  has  the  bearing  re- 
quired. But  it  does  not  pass 
through  the  station  1.  Transfer  it  thither  by  drawing  through 
station  1  a  line  parallel  to  it,  which  will  be  the  course  requked,  its 
proper  length  being  set  off  on  it  from  1  to  2.  Now  suppose  the 
bearing  from  2  to  be  S.  60°  E.  Draw  through  2  a  line  pai-allel 
to  the  fine  passing  through  the  centre  of  the  circle  and  through 
the  opposite  points  S.  60°  E.,  and  N.  60°  W.,  and  it  will  be  the 
line  desired.  On  it  set  off  the  proper  length  from  2  to  8,  and  so 
proceed. 

When  the  plat  is  completed,  the  engraved  sheet  is  laid  on  a 
clean  one,  and  the  stations  "  pricked  through,"  and  the  points  thus 
obtained  on  the  clean  sheet  are  connected  by  straight  lines.  The 
pencilled  plat  is  then  rubbed  off  from  the  engraved  sheet,  which  can 
be  used  for  a  great  number  of  plats. 

If  the  central  circle  be  cut  out,  the  plat,  if  not  too  large,  can  be 
made  directly  on  the  paper  where  it  is  to  remain. 

The  surveyor  can  make  such  a  paper  protractor  for  himself,  with 
great  ease,  by  means  of  the  Table  of  Chords  at  the  end  of  this 
volume,  the  use  of  which  is  explained  in  Art.  (275).  The  engraved 
oues  may  have  shrunk  after  being  printed. 

Such  a  circle  is  sometimes  dra^^m  on  the  map  itself.  This  will 
be  particularly  convenient  if  tlie  bearings  of  any  lines  on  the  map, 


166  COMPASS  SURTEYING.  [part  iir 

not  taken  on  the  ground,  are  likely  to  be  required.     If  ilie  map  be 
very  long,  more  than  one  may  be  needed. 

(273)  Drawing-Board  Protractor.  Such  a  divided  circle,  as 
has  just  been  described,  or  a  circular  protractor,  may  be  placed  on 
a  di'awing  board  near  its  cen;re,  and  so  that  its  0°  and  90°  linea 
are  parallel  to  the  sides  of  the  drawing  board.  Lines  are  then  to 
be  drawn,  through  the  centre  and  opposite  divisions,  by  a  ruler 
long  enough  to  reach  the  edges  of  the  drawing  board,  on  which 
they  are  to  be  cut  in,  and  numbered.  The  drawing  board  thus 
becomes,  in  fact,  a  double  rectangular  protractor.  A  strip  of 
white  paper  may  have  previously  been  pasted  on  the  edges,  or  a 
narrow  strip  of  white  wood  inlaid.  When  this  is  to  be  used  for 
platting,  a  sheet  of  paper  is  put  on  the  board  as  usual,  and  linea 
are  dra-\vn  by  a  ruler  laid  across  the  0°  points  and  the  90°  points, 
and  the  centre  of  the  circle  is  at  once  found,  and  should  be  marked 
(^ .     The  bearings  are  then  platted  as  in  the  last  method. 

(274)  With  a  scale  of  chords.  On  the  plane  scale  contained 
in  cases  of  mathematical  drawing  instruments  will  be  found  a  series 
of  divisions  numbered  from  0  to  90,  and  marked  CH?  or  C* 
This  is  a  scale  of  chords,  and  gives  the  lengths  of  the  chords  of 
any  arc  for  a  radius  equal  ia  length  to  the  chord  of  60°  on  the 
scale.  To  lay  oif  an  angle  with  this  scale,  as  for 
example,  to  draw  a  line  makmg  at  A  an  angle 
of  40°  with  AB,  take,  in  the  dividers,  the  dis- 
tances from  0  to  60  on  the  scale  of  chords  ;  with 
this  for  radius  and  A  for  centre,  describe  an  in- 
definite arc  CD.  Take  the  distance  from  0  to 
40  on  the  same  scale,  and  set  it  off  on  the  arc  as 
a  chord,  from  C  to  some  point  D.  Join  AD,  and 
prolong  it.     BAE  is  the  angle  required. 

The  Sector,  represented  on  page  36,  sujiplies  a  modification  of 
this  method,  sometimes  more  convenient.  On  each  of  its  legs  is 
a  scale  marked  C,  or  CH.  Open  it  at  pleasure  ;  extend  the  com- 
pass from  60  to  60,  one  on  each  leg,  and  with  this  radius  describe 
an  arc.     Then  extend  the  compasses  from  40  to  40,  and  the  dis 


CHAP.  IV.]  Platting  the  Survey.  1G7 

tance  will  be  the  chord  of  40°  to  that  radius.     It  can  be  set  oflf  aa 
above. 

The  smallness  of  the  scale  renders  the  method  with  a  scale  of 
chords  practically  deficient  in  exactness  ;  but  it  serves  to  illustrate 
the  next  and  best  method. 

(27.5)  With  a  Tal)Ie  of  chords.  At  the  end  of  this  volume 
will  be  found  a  Table  of  the  lengths  of  the  chords  of  arcs  for  every 
degree  and  minute  of  the  quadrant,  calculated  for  a  radius  equal 
to  1. 

To  use  it,  take  in  the  compasses  one  inch,  one  foot,  or  any  other 
convenient  distance  (the  longer  the  better)  divided  into  tenths  and 
hundredths,  by  a  diagonal  scale,  or  otherwise.  With  this  as  radius 
describe  an  arc  as  in  the  last  case.  Find  in  the  table  of  chords 
the  lensith  of  the  chord  of  the  desired  angle.  Take  it  from  the 
scale  just  used,  to  the  nearest  decimal  part  which  the  scale  will 
give.  Set  it  off  as  a  chord,  as  in  the  last  figure,  and  join  the  point 
thus  obtained  to  the  starting  point.     This  gives  the  angle  desired. 

The  superiority  of  tliis  method  to  that  which  employs  a  protrac- 
tor, is  due  to  the  greater  precision  with  which  a  straight  line  can 
be  divided  than  can  a  circle. 

A  slight  modification  of  this  method  is  to  take  in  the  compasses 
10  equal  parts  of  any  convenient  length,  inches,  half  inches,  quar- 
ter inches,  or  any  other  at  hand,  and  with  this  radius  describe  an 
arc  as  before,  and  set  off  a  chord  10  times  as  great  as  the  one 
found  in  the  Table,  i.  e.  imagine  the  decimal  pouat  moved  one 
place  to  the  right. 

If  the  radius  be  100  or  1000  equal  parts,  imagine  the  dechnal 
pumt  moved  two,  or  three,  places  to  the  right. 

Whatever  radius  may  be  taken  or  given,  the  product  of  that 
radius  into  a  chord  of  the  Table,  wiU  give  the  chord  for  that  radius. 

This  gives  an  easy  and  exact  ruethod  of  getting  a  right  angle  ; 
by  describing  an  arc  with  a  radius  of  1,  and  setting  off  a  chord 
equal  to  1.4142. 

If  the  angle  to  be  constructed  is  more  than  90°,  construct  on 
the  other  side  of  the  given  point,  upon  the  given  line  prolonged,  an 
angle  equal  to  what  the  given  angle  wants  of  180° ;  i.  e.  ita 
iSuV2)hment,  in  the  language  of  Trigonometry. 


168  COMPASS  SURVEYING.  [pari  iu 

This  same  Table  gives  the  means  of  measuring  any  angle. 
With  the  angular  i^oint  for'  a  centre,  and  1,  or  10,  for  a  radius, 
describe  an  arc.  Measure  the  length  of  the  chord  of  the  arc 
between  the  legs  of  the  angle,  find  this  length  in  the  Table,  and 
the  angle  corresponding  to  it  is  the  one  desired.* 

(276)  With  a  Table  of  natural  sines.  In  the  absence  of  a 
Table  of  chords,  heretofore  rare,  a  table  of  natural  smes,  which  can 
be  found  anywhere,  may  be  used  as  a  less  convenient  substitute. 
Since  the  chord  of  any  angle  equals  twice  the  sine  of  half  the 
ano-le,  divide  the  given  angle  by  two ;  find  in  the  table  the  natural 
sme  of  this  half  angle  ;  double  it,  and  the  product  is  the  chord  of 
the  whole  angle.  This  can  then  be  used  precisely  as  was  the 
chord  in  the  preceding  article. 

An  ingenious  modification  of  this  method  has  been  much  used. 
Describe  an  arc  from  the  given  point  as  centre,  as  in  the  last  two 
articles,  but  with  a  radius  of  5  equal  parts.  Take,  from  a  Table, 
the  length  of  the  natural  sine  of  half  the  given  angle  to  a  radius  of 
10.  Set  off  this  length  as  a  chord  on  the  arc  just  described,  and 
join  the  point  thus  obtained  to  the  given  point. f 

(277)  By  Latitudes  and  Departures.  When  the  Latitudes 
and  Departures  of  a  survey  have  been  obtained  and  corrected,  (aa 
explained  in  Chapter  V),  either  to  test  its  accuracy,  or  to  obtain 
its  content,  they  afford  the  easiest  and  best  means  of  platting  it. 
The  description  of  this  method  will  be  given  in  Art.  (285). 

*  This  Table  will  also  serve  to  find  the  natural  sine,  or  cosine,  of  any  angle 
Multiply  the  given  angle  by  two  ;  find,  in  the  Table,  the  chord  of  this  double 
angle  ;  and  half  of  this  «hord  will  be  the  natural  sine  required  For,  the  chord 
of  any  angle  is  equal  to  twice  the  sine  of  half  the  angle.  To  find  the  cosine,  pro- 
ceed as  above,  with  the  angle  which  added  to  the  given  angle  would  make  90°. 

Another  use  of  this  Table  is  to  insci-ibe  regular  polygons  in  a  circle  by  setting 
off  the  chords  of  the  arcs  which  their  sides  subtend. 

Still  another  use  is  to  divide  an  arc  or  angle  into  any  number  of  equal  parta 
by  setting  off  the  fractional  arc  or  angle.  Fig.  189. 

t  The  reason  of  this  is  apparent  from  the 
figure.  DE  is  the  sine  of  half  the  angle 
BAG,  to  a  radius  of  10  eqiial  paj-ts,  and 
BC  is  the  chord  directed  to  be  set  off,  to  a 
radius  of  5  qual  parts.  BC  is  equal  to  DE  ; 
for  BC  =  2.BF,  by  Trigonometry,  and  DE 

—  2.BF,  by  similar  triangles  ;  hence  BC  =  ' ''      nj 

DE.  — - 


CHAPTER  V. 


LATITUDES  A\D  DEPARTrRES 

(278)  Defiiiitious.  The  Latitude  of  a  point  is  it3  distance 
North  or  South  of  some  "  Parallel  of  Latitude^''  or  line  running 
East  or  West.  The  Longitude  of  a  point  is  its  distance 
East  or  West  of  some  ^'- Meridian^''  or  line  running  North  and 
South.  In  Compass-Surveying,  the  Magnetic  Meridian,  i,  e.  the 
direction  in  which  the  Magnetic  Needle  points,  is  the  line  from 
which  the  Longitudes  of  points  are  measured,  or  reckoned. 

The  distance  which  one  end  of  a  line  is  due  Forth  or  South  of 
the  other  end,  is  called  the  Difference  of  Latitude  of  the  two  enda 
of  the  line  ;  or  its  Nortliing  or  Southing  ;  or  simply  its  Latitude. 

The  distance  which  one  end  of  the  line  is  due  East  or  West  of 
the  otlier,  is  here  called  the  Difference  of  Longitude  of  the  two 
ends  of  the  Hne  ;  or  its  Easting  or  Westing  ;  or  its  Departure. 

Latitudes  and  Departures  are  the  most  usual  terms,  and  will  he 
generally  used  hereafter,  for  the  sake  of  brevity. 

This  subject  may  be  illustrated  geographically,  by  noticing  that 
a  traveller  in  going  from  New- York  to  Buffalo  in  a  straight  line, 
would  go  about  150  miles  due  north,  and  250  miles  due  west. 
These  distances  would  be  the  differences  of  Latitude  and  of  Lonsi- 
tude  between  the  two  places,  or  his  Northing  and  Westing.  Re- 
turning from  Buffalo  to  New- York,  the  same  distances  would  be- 
his  Southinir  and  Eastino;.* 

In  mathematical  language,  the  operation  of  finding  the  Latitude 
and  Longitude  of  a  line  from  its  Bearing  and  Length,  would  be 
called  the  transformation  of  Polar  Co-ordinates  into  Rectangular 
Co-ordinates.  It  consists  in  determining,  by  our  Second  Principle^ 
the  position  of  a  point  which  had  originally  been  determined  by 
the  Third  Principle.     Thus,  in  the  figure,  (which  is  the  same  as 

*  It  should  be  remembeied  that  the  following  discussions  of  the  Latitudes  and 
Ajongitudes  of  tlie  points  of  a  survey  will  not  always  be  fully  applicable  to  those 
of  distant  places,  such  as  the  cities  just  named,  in  consequence  of  the  surface  af 
the  earth  nol  being  a  plane. 


170 


COMPASS  SURVEFIIVG. 


[fart  III 


that  of  Art.  (9)),  the  point  Sis  determin- 
ed bj  the  angle  SAC  and  bj  the  dis- 
tance AS.  It  is  also  determined  bj  the 
distances  AC  and  CS,  measured  at  right 
angles  to  each  other ;  and  then,  supposing  ^^ 
CS  to  run  due  North  and  South,  CS  mil  be  the  Latitude,  and  AC 
the  Departure  of  the  line  AS. 


(279)  Calculation  of  Latitudes  and  Departures. 

be  a  given  hue,  of  which  the  length 
AB,  and  the  bearing  (or  angle,  BAC, 
which  it  makes  with  the  Magnetic 
Meridian),  are  known.  It  is  required 
to  find  the  differeiices  of  Latitude  and 
of  Longitude  between  its  two  extremi- 
ties A  and  B :  that  is,  to  find  AC  and 
CB ;  or,  what  is  the  same  tiling,  BD 
and  DA. 

It  wiU  be  at  once  seen  that  AB  is 
the  hypothenuse  of  a  right-angled  tri- 
angle, m  which  the  "  Latitude"  and  the  "  Departure"  are  the  sides 
about  the  right  angle.  We  therefore  know,  from  the  principles  of 
trigonometry,  that 

AC  =  AB  .  cos.  BAC, 
BC  =  AB  .  sm.  BAC. 

Hence,  to  find  the  Latitude  of  any  course,  multiply  the  natural 
cosine  of  the  bearing  by  the  length  of  the  course ;  and  to  find  the 
Departure  of  any  course,  multiply  the  natural  sine  of  the  bearing 
by  the  length  of  the  course. 

If  the  course  be  Northerly,  the  Latitude  will  be  North,  and 
will  be  marked  with  the  algebraic  sign  -\-,plus,  or  additive;  if 
it  be  Southerly,  the  Latitude  will  be  South,  and  will  be  marked 
with  the  algebraic  sign  — ,  minus,  or  subtractive. 

If  the  course  be  Easterly,  the  Departure  will  be  East,  and 
marked  -f  ,  or  additive ;  if  the  course  be  Westerly,  the  Departuw 
win  be  West,  and  marked  — ,  or  subtractive. 


rHAP.  V.J  Latitudes  and  Departuies.  171 

(280)  Formulas.  The  rules  of  the  preceding  article  may  be 
expressed  thus ; 

Latitude  =  Distance  x  cos.  Bearins:, 
Departure  =  Distance  X  sin.  Bearing.* 
From  these  formulas  may  be  obtained  others,  bj  -svhich,  Nvhen 
any  two  of  the  above  four  things  are  given,  the  remaining  two  can 
be  found. 

When  the  Bearing  and  Latitude  are  given  ; 

Distance  =  — '      '-.     =  Latitude  X   sec.  Bearino', 

COS.  Bearing  o' 

Departure  =  Latitude  x  tang.  Bearing-. 

When  the  Bearing  and  Departure  are  given  ; 
Distance  =   .  "^^"^  '"^     =  Departui^  X  cosec.  Bearing, 

siu.  iJeaiuig  ^  O' 

Latitude  =  Departure  x  cotang.  Bearmg. 

When  the  Distance  and  Latitude  are  given  ; 

r\         -r>        •  Latitiirie 

Cos.  Bearmg  =  7- , 

Departure  =  Latitude  X  tang.  Bearing. 
When  the  Distance  and  Departure  a/re  given  ; 

£-,.       T,        .  Departure 

fern.  Bearmg  =  -7-^ , 

*-•  Distance  ' 

Latitude  =  Departure  X  cotang.  Bearing. 
Wheyi  the  Latitude  and  Departure  are  given  ; 

m  r  T>        •  Dppartiire 

lang.  01  Bearmg  =  --^. — — , 

°  °  Latitude  ' 

Distance  =  Latitude  X  sec.  Bearing. 
Still  more  simply,  any  two  of  these  three  —  Distance,  Latitude 
and  Departure — being  given,  we  have 

Distance  =  v'CLatitude^  +  Departure^) 
Latitude  =  vC^istance^  — Departure^) 
Departure  =  \/(Distance2  —  Latitude^) 

(281)  TraTcrse  Tables.  The  Latitude  and  Departure  of  any 
distance,  for  any  bearing,  could  be  found  by  the  method  given  in 
Art.  (279),  with  the  aid  of  a  table  of  Natural  Smes.     But  to 

*  Whenever  sines,  cosines,  tangents,  &c.,  are  here  named,  they  mean  the  natu 
ral  sines  &c.,  of  an  arc  described  with  a  radius  equal  to  one,  or  to  t:.e  unit  by 
which  the  sines,  &c.,  are  measured. 


172  COMPASS  SrRVEYIXG.  [part  hi 

facilitate  these  calculations,  which  are  of  so  frequent  occurrence 
and  of  so  great  use,  Traverse  Tables  have  been  prepared,  origin- 
ally for  navigators,  (whence  the  name  Traverse),  and  subsequently 
for  surveyors.* 

The  Traverse  Tabie  at  the  end  of  this  volume  gives  the  Latitude 
and  Departure  for  any  bearing,  to  each  quarter  of  a  degree,  and 
for  distances  from  1  to  9. 

To  use  it,  find  in  it  the  number  of  degrees  in  the  bearing,  on 
the  left  hand  side  of  the  page,  if  it  be  less  than  45°,  or  on  the  right 
hand  side  if  it  be  more.  The  numbers  on  the  same  line  running 
across  the  page,!  are  the  Latitudes  and  Departures  for  that  bear- 
ing, and  for  the  respective  distances  —  1,  2,  3,  4,  5,  6,  7,  8,  9, — 
which  are  at  the  top  and  bottom  of  the  page,  and  which  may 
represent  chains,  links,  rods,  feet^  or  any  other  unit.  Thus,  if  the 
bearing  be  15°,  and  the  distance  1,  the  Latitude  would  be  0.966 
and  the  Departure  0.259.  For  the  same  bearing,  but  a  distance 
of  8,  the  Latitude  would  be  7.727,  and  the  Departure  2.071. 

Any  distance,  however  great,  can  have  its  Latitude  and  Depar- 
ture readily  obtained  from  this  table  ;  since,  for  the  same  bearing, 
they  are  directly  proportional  to  the  distance,  because  of  the  simi- 
lar triangles  which  they  form.  Therefore,  to  find  the  Latitude  or 
Departure  for  60,  multiply  that  for  6  by  10,  wdiich  merely  moves 
the  decimal  point  one  place  to  the  right ;  for  500,  multiply  the 
numbers  found  in  the  Table  for  5,  by  100,  i.  e.  move  the  decimal 
point  two  places  to  the  right,  and  so  on.  Merely  moving  the  deci- 
mal point  to  the  right,  one,  two,  or  more  places,  will  therefore 
enable  this  Table  to  give  the  Latitude  and  Departure  for  any  deci- 
mal multiple  of  the  numbers  in  the  Table. 

For  compound  numbers,  such  as  873,  it  is  only  necessary  to 
find  separately  the  Latitudes  and  Departures  of  800,  of  70,  and  of 
3,  and  add  them  together.  But  this  may  be  done,  "with  scarcely 
any  risk  of  error,  by  the  following  sunple  rule. 

*  The  first  Traverse  Talile  for  Surveyors  seems  to  have  been  published  in  1/91, 
by  John  Gale.  The  UKist  extensive  table  is  that  of  Capt.  Boileau,  of  the  British 
army,  being  calculated  for  every  minute  of  bearing,  and  to  five  decimal  jdaces, 
for  distances  from  1  to  10,  The  Table  in  this  volume  was  calculated  for  it,  and 
then  compared  with  the  one  just  mentioned. 

t  In  usin^  this  or  any  similar  Tablp,  lay  a  ruler  across  the  page,  just  above  oi 
below  the  line  to  be  followed  out.     Tliis  is  a  very  valuable  meclianica.  assistance 


CHAP,  v.]  Latitodes  and  Departures.  173 

Write  dovrn  the  Latitude  and  Departure  for  the  first  figure  of 
the  given  number,  as  found  in  the  Table,  neglecting  the  decimal 
point ;  write  under  them  the  Latitude  and  Departure  of  the  second 
figure,  setting  them  one  place  farther  to  the  right ;  under  them 
write  the  Latitude  and  Departure  of  the  third  figure,  setting  them 
one  place  farther  to  the  right,  and  so  proceed  with  all  the  figures 
of  the  given  number.  Add  up  these  Latitudes  and  Departures, 
and  cut  off  the  three  right  hand  figures.  The  remaining  figures 
will  be  the  Latitude  and  Departure  of  the  given  number  in  links, 
or  chains,  or  feet,  or  whatever  unit  it  was  given  in. 

For  example ;  let  the  Latitude  and  Departure  of  a  course  hav 
mg  a  distance  of  873  links,  and  a  bearing  of  20°,  be  required.    In 
the  Table  find  20°,  and  then  take  out  the  Latitude  and  Departure 
for  8,  7  and  3,  in  turn,  placing  them  as  above  directed,  thus : 
Distances.  Latitudes.  Departures. 

800  .7518  2736 

70  6578  2394 

3  2819  1026 

873  820.399  298.566 

Taking  the  nearest  whole  numbers  and  rejecting  the  decimals, 
we  find  the  desired  Latitude  and  Departure  to  be  820  and  299.^* 

When  a  0  occurs  in  the  given  number,  the  next  figure  must  be 
set  two  places  to  the  right,  the  reason  of  which  will  appear  from 
the  following  example,  in  which  the  0  is  treated  like  any  other 
number. 

Given  a  bearing  of  35°,  and  a  distance  of  3048  links. 
Distances.  Latitudes.  Departures, 


3000 

2457 

1721 

000 

0000 

0000 

40 

3277 

2294 

8 

6553 

4589 

3048  2496.323  1748.529 

Here  the  Latitudes  and  Departures  are  2496  and  1749  links. 

*  It  is  frequently  doubtful,  in  many  calcula*ions,  when  the  final  decimal  is  5, 
whether  to  increase  the  preceding  fi^ire  by  one  or  not.  Thus,  43.5  may  be  called 
43  or  44  with  equal  correctness.  It  is  better  in  such  cases  not  to  increase  the 
whole  number,  so  as  to  escape  the  trouble  of  changing  the  original  figure,  and 
the  increased  chance  of  error.  If,  however,  more  than  one  such  a  case  occurs  in 
the  same  column  to  be  added  up,  the  larger  and  smaller  number  should  be  taken 
alternately. 


/:4  COMPASS  SURVEYIi\G.  [part  hi 

When  the  bearing  is  over  45°,  the  names  of  the  columns  must 
be  read  from  the  bottom  of  the  page,  the  Latitude  of  any  bearing, 
as  50°,  being  the  Departure  of  the  complement  of  this  bearing,  or 
40°,  and  the  Departure  of  40°  being  the  Latitude  of  50°,  &c.  The 
reason  of  this  will  be  at  once  seen  on  inspecting  the  last  figure,  (page 
170%  and  imagining  the  East  and  West  line  to  become  a  Meri- 
dian. For,  if  AC  be  the  magnetic  meridian,  as  before,  and  there- 
fore BAG  be  the  bearing  of  the  course  AB,  then  is  AC  the  Lati- 
tude, and  CB  the  Departure  of  that  course.  But  if  AE  be  the 
meridian  and  BAD  (the  complement  of  BAC)  be  the  bearing, 
then  is  AD  (which  is  equal  to  CB)  the  Latitude,  and  DB,  (which 
is  equal  to  AC),  the  Departure. 

As  an  example  of  this,  let  the  bearing  be  63^°,  and  the  distance 
3469  links.     Proceeding  as  before,  we  have 

Distances.  Latitudes.  Departures. 

3000  1350  2679 

400  1800  3572 

60  2701  5358 

9  4051  8037 


3469.  1561.061  3097.817 

The  required  Latitude  and  Departure  are  1561  and  3098  links. 

In  the  few  cases  occurring  in  Com  pass- Surveying,  in  which  the 
bearing  is  recorded  as  somewhere  between  the  fractions  of  a  degree 
given  in  the  Table,  its  Latitude  and  Departure  may  be  found  by 
interpolation.  Thus,  if  the  bearing  be  10 1°,  take  the  half  sum  of 
the  Latitudes  and  Departures  for  10^°  and  10 1°.  If  it  be  10°  20', 
add  one-third  of  the  difference  between  the  Lats.  and  Deps.  for 
10^  and  for  10|°,  to  those  opposite  to  10^°  ;  and  so  in  any  similar 
case. 

The  uses  of  this  table  are  very  varied.  The  prmcipal  applicar 
tions  of  it,  which  will  now  be  explained,  are  to  Testing  the  acew- 
racy  of  surveys ;  to  Supplying  omissions  in  them ;  to  Platting 
them,  and  to  Calculating  their  content.* 

*  The  Traverse  Table  admits  of  many  other  minor  uses.  Thus,  it  may  be  used 
for  solving,  approximately,  any  right-angled  triangle  by  mere  inspection,  the 
bearing  being  taken  for  one  of  the  acute  angles  ;  the  Latitude  being  the  side  ad- 
jacent, the  Departure  the  side  opposite,  and  the  Distan.ce  the  hypothenuse.  Any 
two  of  these  being  given,  the  others  are  given  by  the  Table.  The  Table  will 
therefore  serve  to  show  the  allowance  tu  be  made  in  chaining  on  slopes  (see  Art 


CHAP,  v.] 


Latitudes  and  Departures. 


175 


(282)  Application  to  Testing  a  Survey.  It  is  self-evident, 
that  when  the  surveyor  has  gone  completely  arounl  a  field  or 
farm,  taking  the  bearings  and  distances  of  each  boundary  line,  till 
he  has  got  back  to  the  starting  point,  that  he  has  gone  precisely 
as  far  South  as  North,  and  as  far  West  as  East.  But  the  sum  of 
the  North  Latitudes  tells  how  far  North  he  has  gone,  and  the  sum 
of  the  South  Latitudes  how  far  South  he  has  gone.  Hence  these 
two  sums  will  be  equal  to  each  other,  if  the  survey  has  been  cor- 
rectly made.  In  Uke  manner,  the  sums  of  the  East  and  of  the 
West  Departures  must  also  be  equal  to  each  other. 

We  will  apply  this  prmciple  to  testing  the  accuracy  of  the  sur- 
vey of  which  Fig.  175,  page  151,  is  a  plat.  Prepare  seven 
columns,  and  head  them  as  below.  Find  the  Latitude  and  Depar- 
ture of  each  course  to  the  nearest  link,  and  write  them  in  their 
appropriate  columns.  Add  up  these  columns.  Xhcn  will  the 
difference  between  the  sums  of  the  North  and  South  Latitudes, 
and  between  the  sums  of  the  East  and  West  Departures,  indicate 
the  degree  of  accuracy  of  the  survey. 


STATION. 

BEARING. 

DISTANCE. 

LATITUDE. 

DEPARTURE,     i 

N. 

S. 

E. 

W. 

1 

2 
3 
4 
5 

N.  35°  E. 
N.  831°  E. 
S.  57°  E. 

s.  341°  W. 

N.56i°W. 

2.70 

1.29 

2.22 
3.55 
3.23 

2.21 
.15 

1.78 

1.21 
2.93 

1.55 

1.28 
1.86 

2.00 

2.69 

4.14 

4.14 

4.69 

4.69 

The  entire  work  of  the  above  example  is  given  below. 
35c        16.38  1147  341° 


270. 


57340 


22L140 


1147 
40150 

154.850 


355. 


2480 
4133 
4133 

293.463 


1688 
2814 
2814 

199.754 


(26))  ;  f«r,  look  in  the  column  of  bearings  for  the  slope  of  the  ground,  i.  e.  the 
■ngle  it  makes  witu  the  horizon,  find  the  given  distance,  and  the  Latitude  3orre- 
■ponding  will  be  the  desired  horizontal  measurement,  and  the  difference  between 
it  and  the  Dis'ance  will  be  the  allowance  to  be  made 


176  COMPASS  SURVETIM.  [partiii 

83^0        113                  994             5610      1656  2502 

226                1987                           1104  1668 

1019                8942                           1656  2502 


129.  14.579  128.212  323.       178.296  269.382 


570  1089  1677 

1089  1677 

1089 


The  nearest  link  is  taken 

1677         ^^  ^®  inserted  in  the  Table, 

and  the  remaining  Decimals 


222.  120.879  186.147         are  neglected. 

In  the  preceding  example  the  respective  sums  were  found  to  be 
exactly  equal.  This,  however,  will  rarely  occur  in  an  extensive 
survey.  If  the  difference  be  great,  it  indicates  some  mistake,  and 
the  survey  must  be  repeated  with  greater  care  ;  but  if  the  differ- 
ence  be  small  it  indicates,  not  absolute  errors,  but  only  inaccura- 
cies,  unavoidable  in  surveys  with  the  compass,  and  the  survey  may 
be  accepted. 

How  great  a  difference  in  the  sums  of  the  columns  may  be 
allowed,  as  not  necessitating  a  new  survey,  is  a  dubious  point. 
Some  surveyors  would  admit  a  difference  of  1  link  for  every  3 
chains  in  the  sum  of  the  courses :  others  only  1  link  for  every  10 
chains.  One  writer  puts  the  limit  at  5  Hnks  for  each  station ; 
another  at  25  links  in  a  survey  of  100  acres.  But  every  practical 
surveyor  soon  learns  how  near  to  an  equality  his  instrument  and 
his  skill  will  enable  him  to  come  in  ordinary  cases,  and  can  there- 
fore establish  a  standard  for  himself,  by  which  he  can  judge  whether 
the  difference,  in  any  survey  of  his  own,  is  probably  the  result  of 
an  error,  or  only  of  his  customary  degree  of  inaccuracy,  two  things 
to  be  very  carefully  distinguished.* 

(283)  Application  to  supplying  omissions.  Any  two  omis 
sions  in  the  Field-notes  can  be  suppHed  by  a  proper  use  of  the 
method  of  Latitudes  and  Departures  ;  as  will  be  explained  in  Part 
VII,  which  treats  of  "  Obstacles  to  Measurement,"  under  which 
head  this  subject  most  appropriately  belongs.  But  a  knowledge 
of  the  fact  that  any  two  omissions  can  be  supplied,  should  not  lead 

*  A  French  writer  fixes  the  allowable  diffei'ence  in  chaining  at  1-400  of  leyel 
lines     1-200  of  lines  on  moderate  slopes  ;  1-100  of  lines  on  steep  slopes. 


CHAP  v.] 


Latitudes  and  Departures. 


177 


the  young  surveyor  to  be  negligent  in  making  every  possible  mea- 
surement, since  an  omission  renders  it  necessary  to  assume  all  tlie 
notes  taken  to  be  correct,  the  means  of  testing  them  no  longer 
existing. 


(284)  Balancing  a  Survey.  The  subsequent  applications  of 
this  method  reqmre  the  survey  to  be  previously  Balanced.  This 
operation  consists  in  correcting  the  Latitudes  and  Departures  of 
the  courses,  so  that  their  sums  shall  be  equal,  and  thus  "  balance." 
This  is  usually  done  by  distributing  the  diflferences  of  the  sums 
among  the  courses  in  proportion  to  their  length ;  saying.  As  the 
sum  of  the  lengths  of  all  the  courses  Is  to  the  whole  difference  of 
the  Latitudes,  So  is  the  length  of  each  com-se  To  the  correction 
of  its  Latitude.     A  similar  proportion  corrects  the  Departures.* 

It  is  not  often  necessary  to  make  the  exact  proportion,  as  the 
correction  can  usually  be  made,  with  sufficient  accuracy,  by  noting 
how  much  per  chain  it  should  be,  and  correcting  accordingly. 

In  the  example  given  below,  the  differences  have  purposely  been 
made  considerable.  The  corrected  Latitudes  and  Departures  have 
been  here  inserted  in  four  additional  columns,  but  in  practice  they 
should  be  written  in  red  ink  over  the  original  Latitudes  and 
Departures,  and  the  latter  crossed  out  with  red  ink. 


STA. 

BXARINe, 

DIST. 

LATITUDES. 

dep'tukes. 

CORRECTED 
lATITUDES. 

CORRECTED 
DEPARTURES. 

N.+ 

S.—     E.+ 

W.— 

K+ 

S.— 

E.+ 

W.— 

1 

2 
8 
4 

N.  52°  E. 
S.  29|°  E. 
S.  31f°W. 
N.  61°  W. 

10.63 
4.10 

7.13 

6.54 
3.46 

3.56 
6.54 

8.38 
2.03 

4.05 
6.24 

10.29 

6.58 
3.48 

3.55 
6.51 

8.34 
2.01 

4.08 
6.27 

29.55 

10.00 

10.10 

1041 

10.06 

10.06 

10.35 

10.35 

The  corrections  are  made  by  the  following  proportions ;  the 
nearest  whole  numbers  being  taken : 


For  the  Latitudes. 


For  the  Departures. 


29.55 
29.55 
29.55 
29.55 


10.63 
4.10 
T.69 
7.13 


10 
10 
10 
10 


4 
1 
3 

_2 

10 


29.55 
29.55 
29.55 
29.55 


10.63 
4.10 
7.69 
7.13 


12 
12 
12 

12 


:  4 
:  2 
:  3 

L? 

12 


*  A  demonstration  of  this  principle  was  given  by  Dr.  Bowditch,  in  No.  4  of 
'  The  Analyst."  -tn 


178 


COMPASS  SURVEYING. 


[part  III 


This  rule  is  not  always  to  be  strictly  followed.  If  one  line  of  a 
survey  has  been  measured  over  very  uneven  and  rough  ground,  or 
if  its  bearing  has  been  taken  with  an  indistinct  sight,  while  the 
other  lines  have  been  measured  over  level  and  clear  ground,  it  ia 
probable  that  most  of  the  error  has  occurred  on  that  line,  and  the 
correction  should  be  chiefly  made  on  its  Latitude  and  Departure. 

If  a  shght  change  of  the  bearing  of  a  long  course  will  favor  the 
Balancing,  it  should  be  so  changed,  since  the  compass  is  much 
more  subject  to  error  than  the  chain.  So,  too,  if  shortening  any 
doubtful  line  will  favor  the  Balancing,  it  should  be  done,  since  dis- 
tances are  generally  measured  too  long. 

(285)  Application  to  Platting,  Rule  tliree  columns ;  one  for 
Stations  ;  the  next  for  total  Latitudes  ;  and  the  third  for  total  De- 
partures. Fill  the  last  two  columns  by  beginning  at  any  conven- 
ient station  (the  extreme  East  or  West  is  best)  and  adding  up 
(algebraically)  the  Latitudes  of  the  following  stations,  noticmg 
that  the  Soutb  Lajtitudes  are  sub  tractive.  Do  the  same  for  the 
Departures,  obs'erving  that  the  Westerly  ones  are  also  subtractive. 

Taking  the  example  given  on  page  175,  Art.  (282),  and  begin- 
ning with  Station  1,  the  following  will  be  the  results : 


TOTAL  LATITUDES 

TOTAL  DEPARTURES 

1 

FROM  STATION   1. 

FROM  STATION    1. 

0.00 

0.00 

2 

+  2.21  N. 

+  1.55  E.      i 

3 

+2.36  N. 

+2.83  E.       ' 

4 

+  L15N. 

+4.69  E. 

5 

—1.78  S. 

+  2.69  E. 

1 

0.00 

0.00 

^  Cac'. 


It  will  be  seen  that  the  work  proves  itself,  by  the  total 
Latitudes  and  Departures  for  Station  1,  again  coming  out  equal 
to  zero. 

To  use  this  table,  draw  a  meridian  through  the  point  taken  for 
Station  1,  as  in  the  figure  on  the  following  page.  Set  off,  upward 
from  this,  along  the  meridian,  the  Latitude,  221  links,  to  A,  and 
from  A,  to  the  right  perpendicularly,  set  off  the  Departure,  155 
Imks.*     This  gives  the  point  2.     Joui  1....2.     From  1  again,  set 

*  This  is  most  easily  clone  with  the  aid  of  a  right-angled  triangle,  sliding  one 
of  the  sides  adjacent  to  the  right  angle  along  the  blade  of  the  square,  to  which 
tlie  other  side  will  then  be  perpendicular.  '  .J^ 


CHAP    v.] 


Latitudes  and  Departures. 


179 


Fig.  192. 


off,  upward,  236 
links,  to  B,  and  from 
B,  to  the  right,  per- 
pendicularlj,  set  off 
283  links,  which  will 
fix  the  point  3 .  Join 
2. ...3  ;  and  so  pro- 
ceed, setting  off 
North  Latitudes 
along  the  INIeridian  i 
upwards,  and  South 
Latitudes  along  it 
downwards ;  East 
Departures  perpen- 
dicularly to  the  right, 
and  West  Depar- 
tures perpendicularlj  to  the  left. 

The  advantages  of  this  method  are  its  rapidity,  ease  and  accu- 
racy ;  tlie  impossibility  of  any  error  in  platting  any  one  course 
affecting  the  following  points ;  and  the  certainty  of  the  plat  "  com- 
ing together,"  if  the  Latitudes  and  Departures  have  been  "  Bal- 
eujced." 


CHAPTER  VI. 


CALCULATING  THE  COIVTEINT. 

(2S6)  Methods.  When  a  field  has  been  platted,  by  what 
ever  method  it  may  have  been  surveyed,  its  content  can  be  obtained 
from  its  plat  by  dividing  it  up  into  triangles,  and  measuring  on 
the  plat  their  bases  and  perpendiculars ;  or  by  any  of  the  other 
means  explained  in  Part  I,  Chapter  IV. 

But  these  are  only  approximate  methods ;  their  degree  of  accuracy 
depending  on  the  largeness  of  scale  of  the  plat,  and  the  skill  of  the 
draftsman.  The  invaluable  method  of  Latitudes  and  Departures 
gives  another  means,  perfectly  accurate,  and  not  requiring  the 
previous  preparation  of  a  plat.  It  is  sometimes  called  the  Rectar- 
gular,  or  the  Pennsylvania,  or  Rittenliouse's,  method  of  calculation  * 

(287)  Definitions.  Imagine  a  Meridian  line  to  pass  through 
the  extreme  East  or  West  corner  of  a  field.  According  to  the 
definitions  established  in  Chapter  V,  Art.  (278),  (and  here  reca- 
pitulated for  convenience  of  reference),  the  perpendicular  distance 
of  each  Station  from  that  Meridian,  is  the  Longitude  of  that  Sta- 
tion ;  additive,  or  ^lus,  if  East ;  subtractive,  or  minus,  if  West. 
The  distance  of  the  middle  of  any  line,  such  as  a  side  of  the 
field,  from  the  Meridian,  is  called  the  Longitude  of  that  side.i 
The  difference  of  the  Longitudes  of  the  two  ends  of  a  line  is  called  ' 
the  Departure  of  that  line.  The  difference  of  the  Latitudes  of  the 
two  ends  of  a  line  is  called  the  Latitude  of  the  line. 

*  It  is,  however,  substantially  the  same  as  Mr.  Thomas  Burgh's  "Method  to 
determine  the  areas  of  right  lined  figures  universally,"  published  nearly  a  century 
ago. 

t  The  phrase  "  Meridian  Distance,"  is  generally  used  for  what  is  here  calleri 
'  Longitude";  but  the  analogy  of  "  Difterences  of  Longitude"  with  "  Differenceu 
of  Latitude,"  usually  but  anomalously  united  with  the  word  "  Departure,"  bor- 
rowed from  Navigation,  seems  to  put  beyond  all  question  the  propriety  of  the 
inuovation  here  introduced. 


CHAP,  yi.] 


Calculating  the  Content. 


181 


(288)  Longitudes.  To  give  more  definiteness  to  the  develop 
ment  of  this  subject,  the  figure  in  the  margin  will  be  referred  to, 
and  may  be  considered  to  represent  an}-  space  enclosed  by  straight 
lines. 

Let  NS  be  the  Meridian  passmg  through  the  extreme  Westeily 
Station  of  the  field  ABCDE.   From  Fig.  i93.    • 

the  middle  and  ends  of  each  side 
draw  perpendiculars  to  the  jNIeridi- 
an.  These  perpendiculars  will  be 
the  Longitudes  and  Departui'es  of 
the  respective  sides.  The  Longi- 
tude, FG,  of  the  first  course,  AB, 
is  evidently  equal  to  half  its  Depar- 
ture HB.  The  Longitude,  JK,  of 
the  second  course,  BC,  is  equal  to 
JL  -I-  LM  +  jMK,  or  equal  to  the 
Longitude  of  the  preceding  com-se, 
plus  half  its  Departure,  plus  half 
the  Departure  of  the  course  itself. 
The  Longitude,  YZ,  of  some  other 
course,  as  EA,  taken  anywhere,  is 
equal  to  WX  —  VX  —  UV,  or  equal  to  the  Longitude  of  the  pre- 
ceding  course,  minus  half  its  Departure,  minus  half  the  Departure 
of  the  course  itself,  i.  e.  equal  to  the  Algebraic  sum  of  these  three 
parts,  remembering  that  Westerly  Departures  are  negative,  and 
therefore  to  be  subtracted  when  the  directions  are  to  make  an 
Algebraic  addition. 

To  avoid  fractions,  it  will  be  better  to  double  each  of  the  preced- 
ing expressions.     We  shall  then  have  a 

GENERAL  RULE  FOR  FINDING  DOUBLE  LONGITUDES. 

The  Double  Longitude  of  the  first  course  is  equal  to  its  De- 
parture.   . 

The  Double  Longitude  of  the  second  course  is  equal  to  the 
DmbU  Longitude  of  the  first  course,  plus  the  Departure  of  that 
course,  plus  the  Departure  of  the  second  course. 

The  Double  Longitude  of  the  third  course  is  equal  to  the 
Double  Longitude  of  the  second  course,  plus  the  Departure  of  thai 
^urse^  plus  the  Departure  of  the  course  itself. 


s 


/' 


i^  trU^-LJ^^ 


%^/ 


182 


COMPASS  SURVEYOG. 


f PART  in 


The  Double  Longitude  of  any  course  is  equal  to  tlie  DouhU 
Longitude  of  the  preceding  course^  plus  the  Departure  of  that 
course,  plus  the  Departure  of  the  course  itself* 

The  Double  Longitude  of  the  last  course  (as  well  as  of  the  first) 
is  equal  to  its  Departure.  Its  "coming  out"  so,  when  obtained 
bj  the  above  rule,  proves  the  accuracy  of  the  calculation  of  all  the 
preceding  Double  Longitudes.  v^ 

(289)  Areas,  We  will  now  proceed  to  find  the  Area,  or  Con« 
tent  of  a  field,  by  means  of  the  "  Double  Longitudes"  of  its  sides, 
which  can  be  readily  obtained  by  the  preceding  rule,  whatever  their 

number. 


(290)  Beginning  with  a  three-sided  field,  ABC  in  the  figure,  draw 
a  Jkleridian  through  A,  and  draw  perpendi- 
culars "to  it  as  in  the  last  figure.  It  is 
plain  that  its  content  is  equal  to  the  differ- 
ence of  the  areas  of  the  Trapezoid  DBCE, 
and  of  the  Triangles  ABD  and  ACE. 

The  area  of  the  Triangle  ABD  is  equal 
to  the  product  of  AD  by  half  of  DB,  or  to 
the  product  of  AD  by  FG ;  i.  e.  equal  to 
the  product  of  the  Latitude  of  the  1st  course 
by  its  Longitude. 

The  area  of  the  Trapezoid  DBCE  is  equal 
to  the  product  of  DE  by  half  the  sum  of  DB 
and  CE,  or  by  HJ ;  i.  e.  to  the  product  of 
the  Latitude  of  the  2d  coui'se  by  its  Longitude. 

The  area  of  the  Triangle  ACE  is  equal  to  the  product  of  AE  by 
half  EC,  or  by  KL  ;  i.  e.  to  the  product  of  the  Latitude  of  the;  3d 
course  by  its  Longitude. 

CaUing  the  products  in  which  the  Latitude  was  North,  J^orth 
Products,  and  the  products  in  which  the  Latitude  was  South, 
South  Products,  we  shall  find  the  area  of  the  Trapezoid  to  be  a 
S(yuih  Product,  and  the  areas  of  the  Triangles  to  be  North  Pro- 


•  The  last  course  is  a  "  preceding  course"  to  the  first  course,  as  will  appear  on 
remembering  that  these  two  courses  join  »ach  other  on  the  ground 


SHAP.  VI.] 


Calcnlatln?  the  Content. 


185 


ducts.  The  Difference  of  the  North  Products  and  the  South 
Products  is  therefore  the  desired  area  of  the  three-sided  field  ABC% 
Using  the  Double  Longitudes,  (in  order  to  avoid  fractions),  'm 
each  of  the  preceding  products,  their  difference  v^-Wl  be  the  double 
area  of  the  Triande  ABC. 


(291)  Taking  now  a  four-sided  field,  ABCD  in  the  figure,  and 
drawing  a  Meridian  and  Longitudes  as  be- 
fore, it  is  seen,  on  inspection,  that  its  area 
would  be  obtained  by  taking  the  two  Trian- 
gles, ABE,  ADG,  from  the  figure  EBCDGE, 
or  from  the  sum  of  the  two  Trapezoids  EBCF 
and  FCDG. 

The  area  of  the  Triangle  AEB  will  be 
found,  as  in  the  last  article,  to  be  equal  to 
the  product  of  the  Latitude  of  the  1st  course 
by  its  Longitude.  The  Product  will  be 
North. 

The  area  of  the  Trapezoid  EBCF  will  be 
found  to  equal  the  Latitude  of  the  2d  course 
by  its   Longitude.     The   product   wiU   be 

South. 

The  area  of  the  Trapezoid  FCDG  will  be  found  to  equal  the 
product  of  the  Latitude  of  the  3d  course  by  its  Longitude.  The 
product  will  be  South. 

The  area  of  the  Triangle  ADG  will  be  found  to  equal  the  pro- 
duct of  the  Latitude  of  the  4th  course  by  its  Longitude.  The  pro- 
duct will  be  North. 

The  difference  of  the  North  and  South  products  will  ther^ 
fore  be  the  desired  area  of  the  four-sided  field  ABCD. 

Using  the  Double  Longitude  as  before,  in  each  of  the  preceding 
products,  their  difference  will  be  double  the  area  of  the  field. 

(292)  "Whatever  the  number  or  directions  of  the  sides  of  a  field, 
or  of  any  space  enclosed  by  straight  lines,  its  area  will  always  be 
equal  to  half  of  the  difference  of  the  North  and  South  Products 


184  COMPASS  SrRVEYL\G.  [part  in 

arising  from  multiplying  together  the  Latitude  and  Double  Longi- 
tude of  each  course  or  side. 

We  have  therefore  the  following 

GENERAL  RULE  FOR  FINDING  AREAS. 

1.  Prepare  ten  columns,  headed  as  in  the  example  below,  and 
in  the  first  three  write  the  Stations,  Bearings  and  Distances. 

2.  Find  the  Latitudes  and  Departures  of  each  course,, bi/  the 
Traverse  Table,  as  directed  in  Art.  (281),  placing  them  in  the 
four  following  columns. 

3.  Balance  them,  as  in  Art.  (284),  correcting  them  in  red  ink. 

4.  Find  the  Double  Longitudes,  as  in  Art.  (288),  with  refer- 
ence to  a  Meridian  passing  through  the  extreme  East  or  West 
Station,  and  place  them  in  the  eighth  column. 

5.  Multiply  the  D-ouble  Longitude  of  each  course  by  the  cor- 
rected Latitude  of  that  course,  placing  the  North  Products  in  the 
ninth  column,  and  the  South  Products  im  the  tenth  column. 

6.  Add  up  the  last  two  columns,  subtract  the  smaller  sum  from 
the  larger,  and  divide  the  difference  by  two.  The  quotient  will 
he  the  content  desired. 

(293)  To  find  the  most  Easterly  or  "Westerly  Station  of  a  sur- 
vey, without  a  plat,  it  is  best  to  make  a  rough  hand-sketch  of  the 
survey,  drawing  tiie  lines  in  an  approximation  to  their  true  direc- 
tions, by  drawing  a  North  ani  South,  and  East  and  West  hues, 
and  coNsidering  the  Bearings  as  fractional  parts  of  a  right  angle, 
or  90° ;  a  course  K.  45°  E.  for  example,  being  drawn  about  half 
<ray  between  a  North  and  an  East  direction ;  a  course  N.  28°  W. 
being  not  quite  one-third  of  the  way  around  from  North  to  West ; 
and  so  on,  drawing  them  of  approximately  true  proportional  lengths. 

C294)  Fxamjyle  1,  given  below,  refers  to  the  five-sided  field,  of 
which  a  plat  is  given  in  Fig.  175,  page  151,  and  the  Latitudes  and 
Departures  of  which  were  calculated  in  Art.  (282),  page  175. 
Station  1  is  the  most  Westerly  Station,  and  the  j\Ieridian  will  be 
fupposed  to  pass  through  it.     The  Double  Longitudes  are  best 


CHAP.  TI.] 


CalculatioiT  the  Content. 


185 


found  bj  a  continual  addition  and  subtraction, 
as  in  the  margin,  where  they  are  marked  D.  L. 
The  Double  Longitude  of  the  last  course  comes 
out  equal  tc  its  Departure,  thus  proving  the 
work. 

The  Double  Longitudes  being  thus  obtained, 
are  multiphed  by  the  corresponding  Latitudes, 
and  the  content  of  the  field  obtained  as  directed 
in  the  General  Rule. 

This  example  may  serve  as  a  pattern  for  the 
most  compact  manner  of  arranging  the  work. 


STA. 

J 

2 
3 
4 
5 

-f  1.55  D.  L 
-t-  1.55 
+  1.28 

+  4.38  D.  I. 
+  1.28 
+  1.86 

+  7.-52   D.  I. 
-1-  1.86 
—  2.00 

+  7.38  D.  L. 
—  2.00 

.-  2.69 

+  2.69   D.  L- 

STATION. 

BKARI.VGS. 

EIS- 

TA.VCES. 

L.1T1TUUES. 

dkp'tuue.s 

DUUBLE 
LONGITUDE-. 

OiiUlil.K    1KKA'^.| 

N.+ 

S. — 

E.+ 

W.— 

N.+ 

S.  — 

1 

N.  35°    B. 

2.70 

2.2 1 

1.55 

+  1..55 

3.4255 

2 

N.  83^0  E. 

1.29 

.15 

1.28 

+  4.38 

0.6570 

3 

S.  57°   E. 

2.22 

1.21 

1.86 

+  7.52 

9.0992 

4 

S.  34i°  W. 

3.55 

2.93 

2.00 

+  7.38 

21.6234 

5 

\.  56-1°  W. 

3.23 

1.78 

2.69 

+  2.69 

4.7882 

4.14     4.14 

4.69     4.69 

8.8707 

30.7226 

8.8707 

Uo7itent=lA.  OR.  loP. 


2)21.8519 


Snuare  Cliains,  10.92.=)9 


(295)  The  Meridian  might  equally  well  have 
been  supposed  to  pass  throilgh  the  most  Easterly 
station,  4  in  the  figure.  The  Double  Longitudes 
could  then  have  been  calculated  as  in  the  mar- 
gin. They  will  'of  course  be  all  West,  o;f  minus. 
The  products  being  then  calculated,  the  sum  of 
the  North  products  will  be  found  to  be  29.9625, 
and  of  the  South  products  8.1106,  and  their 
difference  to  be  21.8519,  the  same  result  as  be- 
fore. 


STA. 

4 
5 
1 
i) 

3 

—  2.00    L).  L. 

—  2.00 

—  2.69 

—  6.69   D.  L. 

—  2.69 
+  1  55 

—  7.83   D.  L. 
-f-  1.55 
4-  1.28 

—  5.00   D    L. 
+  1.28 

-}-  1.86 

—  1.86 

(296)  A  number  of  examples,  with  and  without  answers,  will 
now  be  given  as  exercises  for  the  student,  who  should  plat  them 
by  some  of  the  methods  given  in  the  preceding  chapter,  using  each 
of  them  at  least  once.  He  should  then  calculate  their  content  by 
the  method  just  given,  and  check  it,  by  also  calculating  the  area  of 
the  plat  by  some  of  the  Geometrical  or  Instrumental  methods  given 
in  Part  I,  Chapter  IV;  for  no  single  calculation  is  ever  reliable. 


186 


COMPASS  SURVEYIiXG. 


[part  in. 


All  the  examples  (except  the  last)  are  from  the  author's  actual 
surveys. 


Fiff.  196. 


Example  2,  given  below,  is 
also  fullj  worked  out,  as  anoth- 
er pattern  for  the  student,  whc 
need  have  no  difficulty  with  any 
possible  case  if  he  strictly  fol- 
lows the  directions  which  have 
been  given.  The  plat  is  on  a 
scale  of  2  chains  to  1  inch, 
(=1:1584). 


STATION. 

BEARINGS. 

DIS- 
TANCES. 

LATITUDES. 

dep'turks 

double 
longitudes. 

DOUBLE  AREAS. 1 

N.  +  S— 

B.+ 

VV.— 

N.+ 

S. — 

1 

N.  12i°  B. 

2.81 

2.75 

.60 

+  6.56 

18.0400 

o 

N.  76°     W. 

3.20 

.77 

3.11 

+  4.05 

3.1185 

3 

S.  24.i°  W. 

1.14 

1.04 

.47 

+     A7 

.4888 

4 

S.  48°     E. 

1..53 

1.02 

1.14 

+  1.14 

1.1628 

5 

S.  12.^°  E. 

1.12 

l.Of) 

.24 

4-  2..52 

2.7468 

6 

S.  77°     E. 

1.64 

.37 

1.60 

-f   4.36 

1.6132 

3.52  1  3.52|3.58  |  3.58 

21.1585 

6.01i« 

6.0116 

Content  =  OK.  3R.  IP. 


Example  3. 


2)15.1469 


Square  Chains,    7.5734 

Example  4. 


ST  A. 
1 

2 
3 
4 

BEARING. 

DISTANCE. 

N.52°    E. 
S.  29|°  E. 
S.  31|°  AV. 
N.61°   W. 

10.64 
4.09 
7.68 
7.24 

ST  A. 

BEARING. 

DISTANCE. 

1 

2 
3 
4 

S.    21°  W. 

N.  83i°E. 
N.  12°   E. 
N.  47°  W. 

12.41 

5.86 
8.25 
4.24 

Ans.     4A.  3R.  28P. 
Example  5. 


Ans.    4A.  2R.  37P. 
Example  6. 


ST  A. 

BEARING. 

DISTANCE. 

1 

2 
3 
4 
5 

N.  34i°E. 
N.85°   E. 
S.  56f  E. 
S.  341^^  W. 
N.  561°  w. 

2.73 
1.28 
2.20 
3.53 
3.20 

Ans.     lA.  OR.  14P. 


STA. 
1 

BEARING. 

DISTANCE. 

N.  35°    E. 

6.49 

2 

S.  56|°  E. 

14.15 

3 

S.  34°    W. 

5.10 

4 

N.  56°    W. 

5.84 

5 

S.  291°  W. 

2.52 

6 

N.  48^°  W. 

8Y3 

CHAP.  VI.] 


Calculating?  the  Content. 


IS7 


Example  7. 


Example  8. 


^ 
t 


ST  A. 

BEARING. 

DISTANCE. 

1 

S.  21i°  W. 

17.62 

2 

S.  34°   W. 

10.00 

3 

N.  56°   W. 

14.15 

4 

N.  34°    E. 

9.76 

6 

N.  67°    E. 

2.30 

6 

N. -23°    E. 

7.03 

7 

N.  181°  E. 

4.43 

8 

S.  76i°  E. 

12.41 

Example  9. 


ST  A. 

BEARING. 

DISTANCE. 

1 

S.  57°    E. 

5.77 

2 

S.  36|°  W. 

2.25 

3 

S.  39^°  W. 

1.00 

4 

S.  701°  w. 

1.04   1 

5 

N.  68|°  W. 

1.23 

6 

N.  56°    W. 

2.19 

7 

N.  331°  E. 

1.05 

8 

N.  561^'  W. 

1.54 

9 

N.  331°  E. 

3.18 

Ans.     2A.  OR.  32P. 
Example  11. 


V 

^ 


■< 


ST  A. 
1 

BEARING. 

DISTANCE. 

N.  18f  E. 

1.93 

2 

N.  9°      W. 

1.29 

3 

N.  14°    W. 

2.71 

4 

^.  74°    E. 

0.95 

5 

S.  481°  E. 

1.59 

6 

S   14^°E. 

1.14 

7 

S   191°  E. 

2.15 

8 

S.  23|°  W. 

1.22 

9 

S.  5°      W. 

1.40 

10 

S.  30°    W. 

1.02 

11 

S.  811°  w. 

0.69 

12 

N.  321°  w. 

1.98 

STA. 

BEARING. 

DISTANCE. 

1 

S.  651=  E. 

4.98 

2 

S.  58°    E. 

8.56 

3 

s.  141°  W. 

20.69 

4 

S.  47°    W. 

0.60 

5 

s.  571°  w. 

8.98 

6 

N.  56°    W. 

12.90 

7 

N.  34°    E. 

10.00 

8 

N.  211    E. 

17.62 

Examp)le 

10. 

STA. 
1 

BEARING. 

DISTANCE. 

N.  63°  51'  W. 

6.91 

2 

N.  63°  44'  W 

7.26 

3 

N.  69°  35'  W. 

3.34 

4 

N.  77°  50'  W. 

6.54 

5 

N.  31°  24'  E. 

14.38 

6 

N.  31°  18'  E. 

16.81 

7 

S.  68°  55'  E. 

13.64 

8 

S.  68-^  42'  E. 

11.54 

9 

S.  33°  45'  AV. 

31.55 

Ans.     74  Acres. 


Example 

12. 

STA. 

BEARING. 

DISTANCE. 

1 

N.  72|°  E. 

0.88 

2 

S.  201°  E. 

0.22 

3 

S.  63°    E. 

0.75 

4 

N.  51°    E. 

2.35 

5 

N.  44°    E. 

1.10 

6 

N.  251°  AV. 

1.96 

7 

N.  81°    W. 

1.05 

8 

S.  29^    W. 

1.63 

9 

N.  7ir  W. 

0.81 

10 

N.  13|°  W. 

1.17 

11 

N.  63'    W. 

1.28 

12  '      West. 

1.68 

13  iN.  49°    W. 

0.80 

i4;s.  191   E. 

6.20 

i 


jy,   s^  r,LM 


0-, 


188 


COMPASS  SURVEYING. 


[part  ill 


Uxaniple  13.  A  farm  is  described  in  an  old  Deed,  as  bounded 
thus.  Beginning  at  a  pile  of  stones,  and  running  thence  twenty- 
seven  chains  and  seventy  links  South-Easterly  sixty-six  and  a  half 
degrees  to  a  white-oak  stump ;  thence  eleven  chains  and  sixteen 
links  North-Easterly  twen- 
ty and  a  half  degrees  to  a 
hickory  tree ;  thence  two 
chains  and  thirty-five  links 
North-Easterly  thirtj-six 
degrees  to  the  South-East- 
erly corner  of  the  home- 
stead ;  thence  nineteen 
chains  and  thirty-two  links 
North-Easterly  twenty-six 
degrees  to  a  stone  set  in 
the  ground ;  thence  twenty- 
eight  chains  and  eighty  links 
North-Westerly  sixty-six 
degrees  to  a  pine  stump ; 
thence  thirty-three  chains  and  nineteen  links  South-Westerly 
twenty-two  degrees  to  the  place  of  beginning,  containing  ninety-two 
acres,  be  the  same  more  or  less.     Required  the  exact  content. 

y/^  (297)  Mascheroni's  Theorem.  The  surface  of  any  polygon 
is  equal  to  half  the  sum  of  the  products  of  its  sides  (omitting  any 
one  side")  taken  two  and  two,  into  the  sines  of  the  angles  which 
those  sides  make  ivith  each  other. 


Fis.  198. 


Thus,  take  any  polygon,  such  as  the  five- 
sided  one  ui  the  figure.  Express  the  angle  which 
the  directions  of  any  two  sides,  as  AB,  CD,  make 
with  each  other,  thus  (AB^CD).  Then  will 
the  content  of  that  polygon  be,  as  below ; 


=  J [AB  .  BC 

+  AB  .  DE 

f  BC  .  DE 


sin  (AB  A  BC)  -f-  AB  .  CD  .  sin  (AB  A  CD) 
sin  ( AB  A  DE)  -I-  BC  .  CD  sin  (BC  A  CD) 
sin  (BC  A  DE)  -f-  CD  .  DE    sin  (CD  A  DE)] 


CHAP,  vii]        Tarlation  of  the  Magnetic  Needle.  189 

The  demonstration  consists  merely  in  dividing  the  polygon  into 
triangles  by  lines  drawn  from  any  angle,  (as  A)  ;  then  expressing 
the  area  of  each  triangle  by  half  the  product  of  its  base  and  the 
perpendicular  let  fall  upon  it  from  the  above  named  angle ;  and 
finally  separating  the  perpendicular  into  parts  ■which  can  each  bo 
expressed  by  the  product  of  some  one  side  into  the  sine  of  the 
angle  made  by  it  with  another  side.  The  sum  of  these  triangles 
equals  the  polygon. 

The  expressions  are  simplified  by  Ji\-iding  the  proposed  polygon 
into  two  parts  by  a  diagonal,  and  computing  the  area  of  each  part 
separately,  making  the  diagonal  the  side  omitted.* 


CHAPTER  VII. 

THE  VARIATION  OF  THE  MAGNETIC  NEEDLE. 

(298)  Definitions.  The  3Iagnetic  Meridian  is  the  ^[f-J^^ 
direction  indicated  by  the  Magnetic  Needle.  The  True 
Meridian  is  a  true  North  and  South  fine,  which,  if  pro- 
duced, would  pass  through  the  poles  of  the  earth.  The 
Variation^  or  Declination^  of  the  needle  is  the  angle 
which  one  of  these  lines  makes  with  the  other,  f 

In  the  figure,  if  NS  represent  the  direction  of  the  True 
Meridian,  and  N'S'  the  direction  of  the  Magnetic  Meri- 
dian at  any  place,  then  is  the  angle  NAN'  the  Variation 
of  the  Needle  at  that  place. 

(299)  Direction  of  Needle.  The  directions  of  these  two  meri- 
dians do  not  generally  coincide,  but  the  needle  in  most  places 
points  to  the  East  or  to  the  West  of  the  tiue  North,  more  or  less 

"  The  oiiginal  Theorem  is  usually  accredited  to  Lhuillier,  of  Geneva,  who  pub- 
lished it  in  1789.  But  Mascheroni,  the  ingenious  author  of  the  "  Geometry  of 
the  Compasses,"  had  published  it  at  Pavia,  two  years  previously.  The  method 
is  well  de\eloj)ed  in  Prof.  Whitlock's  "  Elements  of  Geometiy.'' 

t  "  Declination''  is  the  more  correct  term,  and  "  Variation"  should  be  reserved 
for  the  change  in  the  Declination  which  will  be  considered  in  the  next  chapter; 
but  custom  iias  established  the  use  of  Variati(m  in  the  sense  of  Declination. 


190 


COMPASS  SURVEYING. 


[part  III 


according  to  the  locality.  Observations  of  the  amount  and  the 
direction  of  this  variation  have  been  made  m  nearly  all  parts  of  the 
world.  In  the  United  States  the  Variation  in  the  Eastern  Statea 
is  Westerly,  and  in  the  Western  States  is  Easterly,  as  vsill  be 
given  in  detail,  after  the  methods  for  determinmg  the  True  Meri- 
dian, and  consequently  the  Variation,  at  any  place,  have  been 
explained. 

TO  DETERMINE  THE  TRUE  MERIDIAN. 

(300)  By  equal  shadows  of  the  Sun.  On  the  South  side  of 
any  level  surface,  erect  an  up- 
right staff,  sho-wn,  in  horizon- 
tal projection,  at  S.  Two  or 
three  hours  before  noon,  nmrk 
the  extremity.  A,  of  its  shadow. 
Describe  an  arc  of  a  circle  with 
S,  the  foot  of  the  staff,  for  cen- 
tre, and  SA,  the  distance  to 
the  extremity  of  the  shadow,  for  radius, 
after  noon  as  it  had  been  before  noon  when  the  first  mark  was 
made,  watch  for  the  moment  when  the  end  of  the  shadow  touches 
the  arc  at  another  point,  B.  Bisect  the  arc  AB  at  N.  Draw  SN, 
and  it  will  be  the  true  meridian,  or  North  and  South  hne  required. 

For  greater  accuracy,  describe  several  arcs  before  hand,  mark 
the  points  in  which  each  of  them  is  touched  by  the  shadow,  bisect 
each,  and  adopt  the  average  of  all.  The  shadow  will  be  better 
defined,  if  a  piece  of  tin  with  a  hole  through  it  be  placed  at  the  top 
of  the  staff,  as  a  bright  spot  will  thus  he  substituted  for  the  less 
definite  shadow.  Nor  need  the  staff  be  vertical,  if  from  its  summit 
a  plumb-line  be  dropped  to  the  ground,  and  the  point  which  this 
strikes  be  adopted  as  the  centre  of  the  arcs. 

This  method  is  a  very  good  approximation,  though  perfectly 
correct  only  at  the  time  of  the  solstices ;  about  June  21st  and 
December  22d.  It  was  employed  by  the  Romans  in  laying  out 
cities. 

To  get  the  Variation,  set  the  compass  at  one  end  of  the  True 
Meridian  line  thus  <)btained,  sight  to  the  other  end  of  it,  and  take 


About  as  many  hours 


CHAP,  vn.]       Variation  of  the  Magnetic  IVecdle. 


191 


the  Bearing  as  of  any  ordinary  line.     The  number  of  degrees  in 
the  reading  will  be  the  desired  variation  of  the  needle. 


-r 


■ig.  201. 
A 


(301)  By  tlie  Nortli  Star,  whicn  in  the  Meridian.     The  North 

Star,  or  Pole  Star,  (called  by  astronomers  Alplia  Ursce  MinoriSy 
or  Polaris),  is  not  situated  precisely  at  the  North  Pole  bf  the 
heavens.  If  it  were,  the  Meridian  could  be  at  once  determined  by 
sighting  to  it,  or  placing  the  eye  at  some  distance  behind  a  plumb- 
line  so  that  this  line  should  hide  the  star.  But  the  North  Star  is 
about  1^^  from  the  Pole.  Twice  in  24  hours,  however,  (more 
precisely  23h.  56m.),  it  is  in  the  Meri- 
dian, being  then  exactly  above  or  below 
the  Pole,  as  at  A  and  C  in  the  figure.  To 
know  when  it  is  so,  is  rendered  easy  by4he 
aid  of  another  star,  easily  identified,  which 
at  these  times  is  almost  exactly  above  or 
below  the  North  Star,  i.  e.  situated  in  the 
same  vertical  plane.  If  then  we  watch  for 
the  moment  at  which  a  suspended  plumb- 
line  will  cover  both  these  stars,  they  will  then  be  in  the  Meridian. 
The  other  star  is  in  the  well  known  constellation  of  t^e  Great 
Bear,  called  also  the  Plough,  or  the  Dipper,  or  Charles's  Wain. 


B* 


C 


Fis.  202. 


•Fig.  203. 


-^1 


S8S5 


S2^ 


fOpUQ 


■♦c 


-\i 


*? 


s 


Two  of  its  five  bright  stars  (the  right-hand  ones  in 


u  -\  r 


Fig.  202)  are 


^ 


known  as  the  "  Pointers,"  from  their  pointing  near  to  the  North      ./^ 


t 


+- 


^ 


>\ 


192  COMPASS  SURVEYIIVG.  [part  hi. 

Star,  thus  assisting  in  finding  it.  The  star  in  the  tail  or  handle, 
nearest  to  the  four  which  form  a  quadrilateral,  is  the  star  which 
comes  to  the  Meridian  at  the  same  time  with  the  North  Star, 
twice  in  24  hours,  as  in  Fig.  202  or  203.  It  is  known  as  Alioth, 
or  Upsilon  Ursce  Majoris* 

To  determine  the  Meridian  by  this  method,  suspend  a  long 
plumb-hne  from  some  elevated  point,  such  as  a  stick  projecting 
from  the  highest  window  of  a  house  suitably  situated.  The  plumb- 
bob  may  pass  into  a  pail  of  water  to  lessen  its  vibrations.  South 
of  this  set  up  the  compass,  at  such  a  distance  from  the  plumb-line 
that  neither  of  the  stars  will  be  seen  above  its  highest  point,  i.  e. 
in  Latitudes  of  40*^  or  50^  not  quite  as  far  from  the  plumb-line  as  it  is 
long.  Or,  instead  of  a  compass,  place  a  board  on  two  stakes,  so 
as  to  form  a  sort  of  bench,  running  East  and  West,  and  on  it  place 
one  of  the  compass-sights,  or  anything  having  a  small  hole  in  it  to 
look  through.  As  the  time  approaches  for  the  North  Star  to  be 
on  the  Meridian  (as  taken  from  the  table  given  below)  place  the 
compass,  or  the  sight,  so  that,  looking  through  it,  the  plumb-hne 
shall  seem  to  cover  or  hide  the  North  Star.  As  the  star  moves 
one  way,  move  the  eye  and  sight  the  other  way,  so  as  to  constantly 
keep  the  star  behind  the  plumb-hne.  At  last  Alioth,  too,  "will  be 
covered  by  the  plumb-hne.  At  that  moment  the  eye  and  the 
plumb-line  are  (approximately)  in  the  Meridian.  Fasten  down  the 
sight  on  the  board  till  morning,  or  with  the  compass  take  the  bear 
ing  at  once,  and  the  reading  is  the  variation.! 

Instead  of  one  plumb-line  and  a  sight,  two  plumb-lines  may  be 
suspended  at  the  end  of  a  horizontal  rod,  turning  on  the  top  of  a 
pole. 

The  line  thus  obtained  points  to  the  East  of  the  true  line  when 
the  North  Star  is  above  Ahoth,  and  vice  versa.  The  North 
Star  is  exactly  in  the  Meridian  about  ^17  minutes  after  it  has  been 
m  the  same  vertical  plane  with  Alioth,  and  may  be  sighted  to  after 
that  interval  of  time,  with  perfect  accuracy. 

*  The  North  Pole  is  very  nearly  at  the  intersection  of  the  line  from  I'olaris  to 
Alioth,  and  a  perpendicular  to  this  line  from  the  small  star  seen  to  the  left  of  it  io 
Fig.  202. 

t  If  a  Transit  or  Theodolite  be  used,  the  cross-hairs  must  be  illuminated  by 
throwing  the  light  of  a  lamp  into  the  telescope  by  its  reflection  from  white  paper 


CHAP.  VII.]       ?arlatJon  of  the  illagnetic  Needle. 


193 


Another  bright  star,  -which  is  on  the  opposite  side  of  the  Pole, 
and  is  known  to  astronomers  as  Gamma  Cassiopeice,  also  comes  on 
the  ]\Ieridian  nearly  at  the  same  time  as  'the  North  Star,  and  -will 
thus  assist  in  determining  its  direction. 

(302)  The  time  at  which  the  North  Star  passes  the  Merirlian 
above  the  Pole,  for  every  10th  day  m  the  year,  is  given  in  the  fol- 
lowing Table,  m  common  clock  time.*  The  upper  transit  is  the 
most  convenient,  since  at  the  other  transit  Alioth  is  too  high  to  be 
conveniently  observed. 


tf^ 


MONTH. 

1st  Day. 

11th  Day. 

21st  Day. 

H.     M. 

H.     M. 

H.     M. 

1 

January, 
Februaiy, 

6  21  p.  M. 
4  18  p.  M. 

5  41  p.  M. 
3  39  p.  M. 

5  02  p.  M. 
3  00  p.  M. 

^ 

March, 

2  28  p.  M. 

1  49  p.  M. 

1  09  p.  M. 

00 

cs 

April, 
May, 

0  26  p.  M. 
10  28  A.  M. 

11    47   A.  M. 

9  49  A.  M. 

11    08   A.  M. 
9    10  A.  M. 

June, 

8  27  A.  M. 

7  48  A.  M. 

7  08  A.  M. 

^4 

July, 
August, 
September, 
October, 

6  29  A.  M. 
4  28  A.  M. 
2  26  A.  M. 
0  28  A.  M. 

5  50  A.  M. 

3  49  A.  M. 

1   47  A.  M. 

11  45  p.  M. 

5    11  A.  M. 

3  09  A.  M. 

1    07  A.  M. 

11  06  p.  M. 

5= 

November, 

10  22  p.  M. 

9  43  p.  M. 

9  04  p.  M. 

December, 

8  24  p.  M. 

7  45  p.  M. 

7  06  p.  M. 

.^\ 


•  To  calculate  the  time  of  the  Noi-th  Star  passing  the  Meridian  at  its  upper  ctil- 
minatioQ  :  Find  in  the  "  American  Almanac,"  (Boston),  or  the  "  Astronomical 
Ephemeris,"  (Washington),  or  the  "  Nautical  Almanac,"  (Loudon),  or  by  interpo- 
lation from  the  data  at  the  end  of  this  note,  the  right  ascension  of  the  star,  and  from 
it  (increased  by  twenty-four  hours  if  uecessai-y  to  render  the  subtraction  possible) 
Bubti'act  the  Right  ascension  of  the  Sun  at  mean  noon,  or  the  sidereal  time  at 
mean  noon,  for  the  given  day,  as  found  in  the  "  Ephemeris  of  the  Sun,"  in  the 
same  Almanacs.  From  the  remainder  subtract  the  acceleration  of  sidereal  oa 
meaiv  time  corresponding  to  this  remainder,  (3m.  56s.  for  24  hours),  and  the  new 
remainder  is  the  required  meaa  solar  time  of  the  upper  passage  of  the  star  acrisss 
the  Meridian,  in  "Astronomical"  reckoning,  the  astronomical  day  beginning  at 
noon  of  the  common  civil  day  of  the  same  date. 

The  right  ascension  of  the  North  Star  for  Jan.  1,  18.50,  is  Ih.  0.5m.  01.48. ;  for 
1860.  Ih.  08m.  02.8s. ;  for  1870,  Ih.  11m.  16.9s. ;  for  1880,  Ih.  14m.  45.1s. ;  for 
1890    Ih.  18m.  29.2s. ;  for  1900,  Ih.  22m.  Sis. 

13 


194  COMPASS  SrRVE¥L\G.  [part  iii. 

To  find  the  time  of  the  star's  passage  of  the  Meridian  for  othei 
days  than  those  given  in  the  Table,  take  from  it  the  time  for  the 
day  most  nearly  prececTmg  that  desired,  and  subtract  from  this 
time  4  minutes  for  each  day  from  the  date  of  the  day  in  the  Table 
to  that  of  the  desired  day ;  or,  more  accurately,  interpolate,  by 
saying :  As  the  number  of  days  between  those  given  in  the  Table 
is  to  the  number  of  days  from  the  next  preceding  day  in  the  Table 
to  the  desired  day,  so  is  the  difference  between  the  times  given  in 
the  Table  for  the  days  next  preceding  and  folloAving  the  desired 
day  to  the  time  to  be  subtracted  from  that  of  the  next  preceding 
day.  The  first  term  of  the  preceding  proportion  is  always  tew, 
except  at  the  end  of  months  having  more  or  less  than  30  days. 
For  example,  let  the  time  of  the  North  Star's  passing  the  Meridian 
on  July  26th  be  required.  From  July  21st  to  August  1st  being 
11  days,  we  have  this  proportion :  11  days  :  5  days  : :  43  minutes  : 
19y\  minutes.  Taking  this  from  5h.  11m.  A.  M.,  we  get  4h. 
51  ^m.  A.  M.  for  the  time  of  passage  required. 

The  North  Star  passes  the  Meridian  later  every  year.  In 
1860,  it  will  pass  the  Meridian  about  two  minutes  later  than  in 
1854 ;  in  1870,  five  minutes,  in  1880,  eight  minutes,  in  1890, 
twelve  minutes,  and  in  1900,  sixteen  minutes,  later  than  in  1854: 
the  year  for  which  the  preceding  table  has  been  calculated. 

The  times  at  which  the  North  Star  passes  the  Meridian  below 
the  Pole,  in  its  lower  Transit,  can  be  found  by  adding  llh.  58m. 
to  the  time  of  the  upper  Transit,  or  by  subtracting  that  interval 
'rom  it.* 

(303)  By  the  North  Star  at  its  extreme  elouffation.    When 

the  North  Star  is  at  its  greatest  ai^parent  angular  distance  East  or 
West  of  the  Pole,  as  at  B  or  D  in  Fig.  201,  it  is  said  to  be  at  ita 
extreme  Eastern,  or  extreme  Western,  Elongation.  If  it  be  observed 
at  either  of  these  times,  the  direction  of  the  Meridian  can  be  easily 


*  The  North  Star,  which  is  now  about  lo  28'  from  the  Pole,  was  12°  distant 
from  it  when  its  place  was  first  recorded.  Its  distance  is  now  diaiinishing  at  the 
rate  of  about  a  third  of  a  minute  in  a  year,  and  will  continue  to  do  so  till  it  ap- 
proaches to  within  half  a  degree,  when  it  will  again  recede.  The  brightest  star 
m  the  Northern  hemisphere,  Alpha  Lyra,  will  be  the  lole  Star  in  about  12,000 
years,  being  then  w-.*jiiu  about  5»  of  the  Pole,  though  now  more  than  51"*  distant 
from  it 


CHAP.  Til.]       Variation  of  the  Magnetic  A'eedle. 


196 


obtained  from  the  observation.  The  great  advantage  of  this  method 
over  the  preceding  is  that  then  the  star's  motion  apparently  ceases 
for  a  short  time. 


(301)  The  following  Table  gives  the 

TDIES  OF  EXTREME  ELONGATIONS  OF  THE  NORTH  STAR.* 


MONTH. 

1st  Day. 

11th  Day. 

21sT  Day. 

EASTERN. 

WESTERN. 

EASTERN. 

WESTERN. 

EASTERN. 

WESTERN.' 

H.    M. 

H.    M. 

H.    M. 

H.    M. 

H.   M. 

H.    M. 

Jan'y, 

0  27  P.M. 

019  a.m. 

1147  a.m. 

1135  p.m. 

1108  a.m. 

10  56  p.m. 

Feb'y, 

10  24  a.m. 

10 13  P.M. 

9  45  A.M. 

9  33  P.M. 

9  06  a.m. 

8  54  p.m. 

March, 

8  34  a.m. 

8  22  p.m. 

7  55  a.m. 

7  43  P.M. 

715  a.m. 

^04  p.m. 

April, 

6  32  a.m. 

6  20  p.m. 

5  53  a.m. 

5  41  P.M. 

514  a.m. 

5  02  P.M. 

May, 

4  34  a.m. 

4  22  p.m. 

3  55  a.m. 

3  43  P.M. 

316  a.m. 

3  04  P.M. 

June, 

2  33  a.m. 

2  21  P.M. 

153  a.m. 

1 42  P.M. 

114  a.m. 

102  p.m. 

July, 

0  35  a.m. 

0  23  P.M. 

1152  p.m. 

1144  a.m. 

11  13  P.M. 

1105  a.m. 

August, 

10  30  p.m. 

10  22a.m. 

9  51  P.M. 

9  43  a.m. 

911p.m. 

9  03  a.m. 

Sept'r, 

8  28  P.M. 

8  20  a.m. 

7  49  p.m. 

7  41a.m. 

7  09  p.m. 

7  01a.m. 

Oct'r, 

6  30  P.M. 

6  22  a.m. 

5  51  P.M. 

5  43  a.m. 

5 12  P.M. 

5  04a.m. 

Nov'r,     4  28  p.m. 

4  21a.m. 

3  49  P.M. 

3  41a.m. 

3 10  P.M. 

3  02  a.m. 

Dec'r,  1  2  30  p.m. 

2  22  a.m. 

151p.m. 

143  a.m. 

112  p.m. 

104  A.M. 

The  Eastern  Elongations  from  October  to  March,  and  the  West- 
ern Elongations  from  April  to  September,  occurring  in  the  day 
time,  they  will  generally  not  be  visible  except  with  the  aid  of  a 
powerful  telescope. 

*  To  calculate  the  times  of  the  greatest  elongation  of  the  North  Star :  Find  in 
one  of  the  Almanacs  before  referred  to,  or  from  the  data  below,  its  Polar  dis- 
tance at  the  given  time.  Add*the  logarithm  of  its  tangent  to  the  logarithm  of  the 
tangent  of  the  Latitude  of  the  place,  and  the  sum  will  be  the  logarithm  of  the 
cosine  cf  the  Hour  angle  before  or  after  the  culmination.  Reduce  the  space  to 
time;  correct  for  sidereal  acceleration  (3m.  56s.  for  24  hours)  and  subtract  the 
result  from  the  time  of  the  star's  passing  the  meridian  on  that  day,  to  get  the  time 
of  the  Eastern  elongation,  or  add  it  to  get  the  Western. 

The  Polar  distance  of  the  North  Star,  for  Tan.  1,  1850,  is  1°  29'  25";  for  1860, 
1»26'  12'.7;  for  1870,  1»  23"  01";  for  1880,  1°  19  50". -1;  for  1890,  1°  16'  40" 7; 
for  1900,  P  13'  32".2. 


-1-1     '^-^C-t^o. 


196  C03IPASS  SURTETDG.  [part  in 

The  preceding  Table  was  calculated  for  Latitude  40".  Th« 
Time  at  wMch  the  Elongations  occur  vary  slightly  for  other  Lati- 
tudes. In  Latitude  50°,  the  Eastern  Elongations  occur  about  2 
minutes  later  and  the  Western  Elongations  about  2  minutes  earlier 
than  the  times  in  the  Table.  In  Latitude  26°,  precisely  the 
reverse  takes  place. 

The  Times  of  Elongation  are  continually,  though  slowly,  becom- 
ing later.  The  preceding  Table  was  calculated  for  July  1st,  1854. 
In  1860,  the  times  will  be  nearly  2  minutes  later;  and  in  1900, 
the  Eastern  Elongations  wiU  be  about  15  minutes,  and  the  Western 
Elongations  17  minutes  later  than  in  1854. 

(305)  Observations.  Knowing  from  the  precedmg  Table  the 
hour  and  minute  of  the  extreme  Elongation  on  any  day,  a  little 
before  that  time  suspend  a  plumb-line,  precisely  as  in  Art.  (301), 
and  place  yourself  south  of  it  as  there  directed.  As  the  North 
Star  moves  one  way,  move  your  eye  the  other,  so  that  the  plumb- 
line  shall  continually  seem  to  cover  the  star.  At  last  the  star  will 
appear  to  stop  moving  for  a  time,  and  then  begin  to  move  back- 
wards. Fix  the  sight  on  the  board  (or  the  compass,  &c.)  in  the 
position  in  which  it  was  when  the  star  ceased  moving ;  for  the  star 
was  then  at  its  extreme  apparent  Elongation,  East  or  West,  as  the 
case  may  be. 

(306)  Azimuths.     The  angle  which  the  line  from  the  eye  to  the 

plumb-line,  makes  with  the  True  Meridian  (i.  e.  the  angle  between 
the  meridian  plane  and  the  vertical  plane  passing  through  the  eye 
and  the  star)  is  called  the  Azimuth  of  the  Star.  It  is  given  in  the 
following  Taole  for  diflferent  Latitudes,  and  for  a  number  of  years  to 
come.  For  the  intermediate  Latitudes,  it  can  be  obtained  by  a 
simple  proportion,  similar  to  that  explained  in  detail  in  Art.  (302).* 

*  To  calculate  this  Azimuth  :  From  the  logai-ithm  of  the  sine  of  the  Polar  dis 
tance  of  the  star,  subtract  the  logarithm  of  the  cosine  of  the  Latitude  of  the  place  ; 
the  remainder  will  be  the  logarithm  of  the  sine  of  the  angle  required  The  Po 
•ar  distance  can  be  obtained  as  directed  in  the  last  note 


_    ^-7 


L   /  .r 


CHAP  VII.]       Tarktion  of  ihe  illagnetic  Needle. 


197 


AZIMUTHS  OF  THE  NORTH  STAR. 

f  Ladtudee 

1854 

1855 

1856 

1857 

1858 

1859  1860 

1870 

50^ 

2°  16|' 

2°  161' 

2°  16' 

23  151' 

23  15' 

23  141' 

23  141 ' 

23  091' 

490 

20  14' 

20  131' 

2°  13^' 

23  12|' 

23  121 ' 

23  12' 

23  111' 

20  061' 

480 

20  111' 

2'3  11' 

23  101' 

23  10^' 

23  09|' 

2'3  091' 

23  09' 

2^04' 

470 

20  09' 

2=  081' 

23  08' 

23  07|' 

23  071' 

23  06|' 

23  0~6i' 

23  011' 

.463 

2^061' 

20  06^' 

23  05|' 

2°  051' 

2^05' 

23  041 ' 

20  041' 

13  591' 

453 

23  O41' 

2^04' 

23  031' 

2°  03^' 

23  02|' 

23  02^' 

23  02' 

l°57i' 

440 

2°  02-1' 

20  02' 

23  Olf 

23  Oil' 

23  01' 

23  001' 

23  00' 

13  55 1' 

43°* 

2°  001' 

2°  00' 

10  591' 

13  59' 

13  58|' 

1°  581 ' 

1^58' 

I3531' 

4  20 

l°58i' 

1058' 

10  5T-1' 

1^57^' 

1°  56|' 

13  56-1' 

1^56' 

1°  51|' 

410 

1°  56|' 

1°  561' 

1°  55|' 

13  55 1' 

13  55' 

13  54-1' 

1°  54^' 

13  50' 

40^ 

1°55' 

lo64i' 

13  54' 

13  53|' 

10  531 ' 

13  53' 

1°  521' 

13  481 ' 

390 

10  53i' 

1°  52|' 

13  521' 

13  52' 

1°  51|' 

13  51^' 

I05I' 

1°  46|' 

38^ 

1°  51|' 

1°  511' 

13  51' 

13  50-1' 

13  50' 

1°  49|' 

1°  49i' 

13  451' 

370 

P  501' 

1°  49|' 

10  491' 

1°  49' 

1°  48|' 

13  481' 

P  48'  • 

13  44' 

36^ 

1°  48|' 

1=  481' 

13  48' 

1°47|' 

P  411' 

1047/ 

13  461' 

13  42|' 

35° 

I0471' 

1047/ 

13  46|' 

13  461' 

13  46' 

1°  451' 

13  45I' 

1-  41 1' 

340 

1°  46^' 

1°  45|' 

13  451' 

13  45' 

13  44|' 
I0431' 

1°  441' 

1^44' 

1040;' 

330 

19  45' 

l°44i' 

13  441 ' 

13  43|' 
1°  42|' 

10  43' 

I0421' 

13  39' 

32° 

1°44' 

1°  43 1' 

1<3  43' 

13  421 ' 

13  42' 

1°  41|' 

10  38' 

31° 

10  423  / 
10  41^' 

1°  421' 

1°42' 

13  411' 

13  41' 

10  40|' 

13  401' 

13  37' 

30'^ 

P  411' 

1044/ 

1°  401' 

1°  40^' 

13  40' 

13  39I' 

13  36' 

(307)  Setting  out  a  Meridian.  When  two  points  in  the 
tion  of  the  North  Star  at  its  extreme  elongation  have  been  F 
obtained,  as  in  Art.  (305),  the  True  Meridian  can  be 
found  thus.  Let  A  and  B  be  the  Wo  points.  Multiply  the 
natural  tangent  of  the  Azimuth  given  in  the  Table,  by  the 
distance  AB.  The  product  will  be  the  length  of  a  Ime 
vrhich  is  to  be  set  off  from  B,  perpendicular  to  AB,  to 
some  point  C.  A  and  C  will  then  be  pomts  in  the  True 
Meridian.     This  operation  may  be  postponed  till  mornmg. 

If  the  directions  of  both  the  extreme  Eastern  and  extreme 
Western  elongations  be  set  out,  the  Ime  lying  midway 
between  them  will  be  the  True  Meridian. 


du'ec- 


Cn'B 


^7' 


198  COMPASS  SURVEYING.  [part  iit 

(308)  Determining-  tlie  Tariation.  The  variation  would  of 
course  be  given  by  taking  the  Bearing  of  the  IMeridian  thus 
obtained,  but  !.'•  can  also  be  determined  by  taking  the  Bearing  of 
the  star  at  the  time  of  the  extreme  elongation,  and  applying  the 
following  rules. 

When  the  Azimuth  of  the  star  and  its  magnetic  bearing  are  one 
East  and  the  other  West,  the  sum  of  the  two  is  the  INIagnetic  Vari- 
tioa,  which  is  of  the  same  name  as  the  Azimuth  ;  i.  e.  East,  if  that 
be  East,  and  West,  if  it  be  West. 

When  the  Azimuth  of  the  star  and  its  Magnetic  Bearing  are 
both  East,  or  both  West,  their  difference  is  the  Variation,  which 
will  be  of  the  same  name  as  the  Azimuth  and  BeaiiLg,  if  the  Azi- 
muth be  the  greater  of  the  two,  or  of  the  contrary  name  if  the 
Azimuth  be  the  smaller.  F'?-  205. 

All  these  cases  are  presented  together  in  the  ^    t    |    f     -^ 
figure,  in  which  P  is  the  North  Pole  ;  Z  the  place 
of  the  observer;  ZP  the  True  Meridian;  S  the 
star  at  its  greatest  Eastern  elongation ;  and  ZN, 
ZN',ZN",  various  supposed  directions  of  the  needle. 

Call  the  Azimuth  of  the  star,  i.  e.  the  angle 
rZS,  2°  East. 

Suppose  the  needle  to  point  to  N,  and  the  Bear- 
mg  of  the  star,  i.  e.  SZN,  to  be  5°  West  of  Mag- 
netic North.  The  variation  PZN  will  evidently  be 
7°  East  of  true  North.  Z 

Suppose  the  needle  to  point  to  N',  and  the  bearing  of  the  star, 
i.  e.  N'ZS,  to  be  1|°  East  of  Magnetic  North.  The  Variation 
will  be  1°  East  of  true  North,  and  of  the  same  name  as  the  Azimuth, 
because  that  is  greater  than  the  bearing. 

Suppose  the  needle  to  point  to  N"  and  the  bearing  of  the  star, 
i.  e.  N"ZS,  to  be  10°  East  of  Magnetic  North.  The  Variation 
will  be  8°  West  of  true  North,  of  the  contrary  name  to  the  Azimuth, 
because  that  is  the  smaller  of  the  two.* 

*  Algebraically,  always  subtract  the  Bearing  from  the  Azimuth,  and  give  the  re- 
mainder its  proper  resuhing  algebraic  sign.  It  will  be  the  Variation;  East  if  plua, 
and  West,  if  minus.  Thus  in  the  first  case  above,  the  Variation  =  +  2°  — 
(—  5°)  =  +  7°  =  7°  East.  In  the  second  case,  the  Variation  =  ^-  2°  —  (  -1-  li") 
■=  -I-  1°  =  1°  East.  In  the  third  case,  the  Variation  =  +  2°  —  <■+  10°^  »» 
_  8°  =  8°  West. 


CHAP.  VII. ]        Variation  of  the  Magnetic  Needle.  199 

If  the  star  was  on  the  other  side  of  the  Pole,  the  rules  would 
apply  likewise. 

(309)  Other  Methods.  Many  other  methods  of  determming  the 
true  Meridian  are  employed ;  such  as  by  equal  altitudes  and  azi- 
muths of  the  sun,  or  of  a  star  ;  by  one  azimuth,  knowing  the  time  ; 
by  observations  of  circumpolar  stars  at  equal  times  before  and  after 
their  culmination,  or  before  and  after  their  greatest  elongation,  &c 

All  these  methods  however  require  some  degree  of  astronomical 
knowledge ;  and  those  which  have  been  exp.kined  are  abmidantly 
sufficient  for  all  the  purposes  of  the  ordinary  Land-Surveyor. 

"  Burt's  Solar  Compass "  is  an  instrument  by  which,  "  when 
adjusted  for  the  Sun's  declination,  and  the  Latitude  of  the  place, 
the  azimuth  of  any  hue  from  the  true  North  and  South  can  be  read 
off,  and  the  difference  between  it  and  the  Bearing  by  the  compass 
will  then  be  the  variation." 

(310)  Magnetic  variation  in  the  United  States.  The  vari- 
ation of  the  Magnetic  needle  in  any  part  of  the  United  States, 
can  be  approximately  obtained  by  mere  inspection  of  the  map 
at  the  beginning  of  this  volume.*  Through  all  the  places  at 
which  the  needle  in  1850,f  pointed  to  the  true  North,  a  line  is 
drawn  on  the  map,  and  called  the  Line  of  no  Variation.  It  will 
be  seen  to  be  nearly  straight,  and  to  pass  in  a  N.N.W.  direction 
from  a  little  west  of  Cape  Hatteras,  N.  C.  through  the  middle  of 
Yirginia,  about  midway  between  Cleveland,  (Ohio),  and  Erie, 
(Pa.),  and  through  the  middle  of  Lake  Erie  and  Lake  Hm'on.  If 
followed  South-Easterxy  it  would  be  found  to  touch  the  most 
Easterly  point  of  South  America.  It  is  now  slowly  moving 
Westward. 

At  all  places  situated  to  the  East  of  this  line  (including  the 
New-England  States,  New-York,New-J ersey,  Delaware,  Maiy- 
land,  nearly  all  of  Pennsylvania,  and  the  EasteiTi  half  of  Yirginia 
and  North  Carolina)  the  Variation  is  "Westerly,  i.  e.  the  north  end 
of  the  needle  points  to  the  west  of  the  true  North.   At  all  places 

•  Copied  (bj  permission)  from  one  prepared  in  1S56,  by  Prof.  A.  D.  Bache,  Supt.  U. 
8.  Coast  Survey,  from  the  U.  S.  C.  S.  Observations.  The  dotted  portions  of  the  line* 
are  interpolations  due  to  the  kindness  of  J.  E.  Hilgard,  Assist.  U.  S.  Coast  Survey. 

t  A  gradual  changre  in  the  Variation  is  going  on  from  year  to  year,  as  will  be  ex- 
plained in  the  next  Chapter. 


200  eOMPASS  SURVEYING.  [part  u 

situated  to  the  West  of  this  line  (including  the  Western  and  South- 
ern States)  the  Variation  is  easterly,  i.  e.  the  North  end  of  the 
needle  points  to  the  East  of  the  true  North.  This  variation  in- 
creases in  proportion  to  the  distance  of  the  place  on  either  side  of 
tiie  line  of  no  variation,  reaching  21°  of  Easterly  Variation  in  Ore- 
gon, and  18°  of  Westerly  Variation  in  Maine. 

Lines  of  equal  Variation  are  lines  drawn  through  all  the  places 
which  have  the  same  variation.  On  the  map  they  are  drawn  for 
each  degree.  All  the  places  situated  on  the  line  marked  1°,  East 
or  West,  have  1°  Variation ;  those  on  the  2°  line,  have  2°  Varia- 
tion, &c.  The  variation  at  the  intermediate  places  can  be  approxi- 
mately estimated  by  the  eye.     These  lines  all  refer  to  1840. 

The  Imes  of  equal  Variation,  if  continued  Northward,  would  all 
meet  in  a  certain  point  called  the  3Iagnetic  Pole,  and  situated  in 
the  neighborhood  of  96°  West  Longitude  from  Greenwich,  and  70° 
of  North  Latitude.     Towards  this  pole  the  needle  tends  to  point. 

Another  Magnetic  pole  is  found  in  the  Southern  hemisphere  ; 
but  the  farther  development  of  this  subject  belongs  to  a  treatise  on 
Natural  Philosophy. 

The  Variation  on  the  Pacific  slope  of  this  country  has  been  very 

imperfectly  ascertained.     A  few  leading  points  are  as  below. 

Cahfornia ;       Point  Conception,  Sept.  1850,  13°  49^'  E. 

Point  Penos,  Monterey,     Feb.   1851, 14°  58'    E. 

Presidio,  San  Francisco,     Feb.    1852, 15°  27'    E. 

San  Diego,  Mar.  1851,  12°  29'    E. 

Oregon ;  Cape  Disappointment,        July,  1851,  20°  45'    E. 

Ewing  Harbor,  Nov.  1861,  18°  29'    E. 

Wash.  Ter'y.  Scarboro'  Harbor,  Aug.  1852,  21°  30'    E. 

•A- 

(311)  To  correct  Magnetic  Bearings.  The  Variation  at  any 
place  and  time  bemg  known,  the  Magnetic  Bearings  taken  there 
and  then,  may  be  reduced  to  their  true  Bearings,  by  these  Rules. 

Rule  1.  When  the  Variation  is  West,  as  it  is  in  the  North- 
Eastern  States,  the  true  Bearing  will  be  the  sum  of  the  Variation 
and  a  Bearing  which  is  North  and  West,  or  South  and  East ;  and 
the  difference  of  the  Variation  and  a  Bearing  which  is  North  and 
East,  or  South  and  West.     To  apply  this  to  the  cardinal  points,  a 


CHAP.  vii.J       Variation  of  the  Magnetic  IVeedie. 


201 


Fig 


,  206. 

A 


North  Beaiing  must  be  called  N.  0°  West,  an  East  Bearmg  N. 
90^  B.,  a  South  Bearing  S.  0°  E.,  and  a  West  Bearing  S.  90^  W. ; 
counting  around  from  N'  to  N,  in  the  figure,  and  so  onward,  '■'  \Yith 
tlie  Sun." 

Tlie  reasons  for  these  corrections 
are  apparent  from  the  Figure,  in  which 
the  dotted  lines  and  the  accented  let- 
ters represent  the  direction  of  the  nee- 
dle, and  the  full  lines  and  the  unac- 
cented letters  represent  the  true  North  -vv- 
and  South  and  East  and  West  lines. 

When  the  sum  of  the  Variation  and 
the  Bearing  is  directed  to4he  taken, 
land  comes  to  more  than  90°j  the  sup- 
plement of  the  sum  Ls  to  ha  taken,  and  the  first  letter  changedv- 
When  the  difference  is  directed  to  be  taken,  and  the  Variation  ia 
greater^thaii^  tiie^earing,  the  last  letter  must  be  changed.  A 
diagram  of  the  case  will  remove  all  doubts.  Examples  of  all  these 
cases  are  given  below  for  ajVariation  of  S°  West. 


tV- 


^ 


S' 


MAGNETIC 
BEARING. 

TRUE 
BEARING. 

MAGNETIC 
BEARING. 

TRUE 
BEARING. 

Noi'tii. 
N.    l^E. 
N.  40°  E. 

East. 
S.  50°  E. 
1      S.  89°  E. 

N..,.il°_W.- -- 
N.    7°W. 
N.  32°  E. 
N.  82°  E. 
S.  58°  E. 
N.  83°  E. 

South. 
S.    2^W. 
S.  60°  W. 

West. 
N.  70°  W. 
N.  83°  W. 

S.      8°E. 

s.  e-'jE.  ■ 

S:52"' W. 
S.  82°  W. 
N.  78°  W. 
S.  89°  W. 

Xx' 


Rule  2.  When  the  Variation  is 
East,  as  in  the  Western  and  Southern 
States,  the  preceding  directions  must 
be  exactly  reversed ;  i.  e.  the  true 
Bearing  will  be  the  difference  of  the 
Variation  and  a  Bearing  which  is  ^^' 
North  and  West,  or  South  and  East ; 
and  the  sum  of  the  Variation  and  a 
Bearing  which  is  North  and  East,  or 
South  and  West.     A  North  Bearins; 


Fig.  207. 

N 


v.< 


ir 


— B 


202 


COMPASS  SURVETOG. 


[part    III 


must  be  called  N.  0°  E.,  a  West  Bearing  N.  90^  W.,  a  South 
Bearing  S.  0°  "VV.,  and  an  East  Bearing  S.  90°  E.,  counting  from 
N'  to  N,  and  so  onward,  "against  the  sun."  The  reasons  foi 
these  rules  are  seen  in  the  Figure.  Examples  are  given  below,  foi 
a  Variation  of  5°  E. 


MAGJfETIC 

TRUE 

MAGNETIC 

TRUE 

BEARING. 

BEARING. 

BEARING. 

BEARING. 

North. 

Jf«    5tJL.... 

South. 

S.      5°W. 

N.  40°  E. 

N.  45°  E. 

S.  60°  W. 

S.  65°  W. 

N.  89°  E. 

S.  86°  E. 

S.  87°  W. 

N.  88°  W. 

East. 

S.  85°  E. 

West. 

N.  85°  W. 

S.    1°E. 

S.    4°W. 

N.  70°  W. 

N.  65°  W. 

S.  50°  E. 

S.  45°  E. 

N.    2°W. 

N.    3°  E. 

(312)  To  survey  a  line  with  true  Bearings.  The  compass 
may  be  set,  or  adjusted,  bj  means  of  the  Vernier,  (noticed  in  Arts. 
(229)  and  (237),  and  shown  in  Fig.  148,  page  126)  according  to 
the  Variation  in  any  place,  so  that  the  Bearings  of  any  lines  then 
taken  with  it  will  be  their  true  Bearings.  To  effect  this,  turn  aside 
the  compass  plate,  by  means  of  the  Tangent  Screw  which  movea 
the  Vernier,  a  number  of  degrees  equal  to  the  Variation,  moving 
the  S.  end  of  the  Compass-box  to  the  right,  (the  North  end  being 
supposed  to  go  ahead)  if  the  Variation  be  Westerly,  and  vice  versa; 
for  that  moves  the  North  end  of  the  Compass-box  in  the  contrary 
direction,  and  thus  makes  a  line  which  before  was  N.  by  the  nee- 
dle, now  read,  as  it  should  truly.  North,  so  many  degrees.  West 
if  the  Variation  was  West;  and  similarly  in  the  reverse  case. 


OMAP.  VIII.  ^  203 


CHAPTER  VIII. 


CHANGES  IN  THE  FARIATION. 

(313)  The  Changes  in  the  Variation  are  of  more  practical 
importance  than  its  absolute  amount.  They  are  of  four  kinds : 
Irregular,  Diurnal,  Annual  and  Secular. 

(314)  Irregular  chang'eSi  The  needle  is  subject  to  sudden 
and  violent  changes,  which  have  no  known  law.  They  are  some- 
times coincident  with  a  thunder  storm,  or  an  Aurora  Borealis, 
(during  which,  changes  of  nearly  1°  in  one  minute,  2i°  in  eight 
minutes,  and  10°  in  one  night,  have  been  observed),  but  often 
have  no  apparent  cause,  except  an  otherwise  invisible  "  Magnetic 
Storm." 

(315)  The  Diurnal  change.  On  continuing  observations  of 
the  direction  of  the  needle  throughout  an  entire  day,  it  will 
be  found,  in  the  Northern  Hemisphere,  that  the  North  end 
of  the  needle  moves  Westward  from  about  8  A.  M.  till  about  2 
P.  M.  over  an  arc  of  from  10'  to  15',  and  then  gradually  returns 
to  its  former  position.*  In  the  Southern  Hemisphere,  the  direction 
of  this  motion  is  reversed.  The  period  of  this  change  being  a  day, 
it  is  called  the  Diurnal  Variation.  Its  eflFect  on  the  permanent 
Variation  is  necessarily  to  cause  it,  in  places  where  it  is  West,  to 
attain  its  maximum  at  about  2  P.  M.,  and  its  minimum  at  about  8 
A.  M. ;  and  the  reverse  where  the  Variation  is  East. 

This  Diurnal  change  adds  a  new  element  to  the  inaccuracies  of 
the  compass ;  since  the  Bearings  of  any  line  taken  on  the  same 
day,  at  a  few  hours  interval,  might  vary  a  quarter  of  a  degree, 
which  would  cause  a  deviation  of  the  end  of  the  line,  amountnig  to 
nearly  half  a  link  at  the  end  of  a  chain,  and  to  35  links,  or  23 
feet,  at  the  end  of  a  mile.  The  hour  of  the  day  at  which  any 
important  Bearing  is  taken  should  therefore  be  noted. 

•  A  similar  but  smaller  movement  lakes  place  during  the  night 


204  CO.IIPASS  SURVEnXG.  [part  m. 

(316)  The  Annual  change.  If  the  observations  be  continued 
throughout  an  entire  year,  it  will  be  found  that  the  Diurnal  changes 
vary  with  the  seasons,  being  about  twice  as  great  in  Summer  as  in 
Winter.  The  period  of  this  change  being  a  year,  it  is  called  the 
Annual  Variation. 

(317)  The  Secular  change.  When  accurate  observations  on 
the  Variation  of  the  needle  in  the  same  place  are  continued  for 
several  years,  it  is  found  that  there  is  a  continual  and  tolerably 
regular  increase  or  decrease  of  the  Variation,  continuing  to  pro- 
ceed in  the  same  direction  for  so  long  a  period,  that  it  may  be 
called  the  Secular  change  of  Variation.* 

The  most  ancient  observations  are  those  taken  in  Paris.  In  the 
year  1541  the  needle  pointed  7°  East  of  North ;  in  1580  the  Vari- 
ation had  increased  to  11^'  East,  being  its  maximum ;  the  needle 
then  began  to  move  Westward,  and  in  1666,  it  had  returned  to  the 
JNIeridian ;  the  Variation  then  became  West,  and  continued  to 
increase  till  in  1814  it  attained  its  maximum,  being  22°  34'  West 
of  North.  It  is  now  decreasing,  and  in  1853  was  20°  17'  W.  In 
London,  the  Variation  in  1576  was  11°  15'  E. ;  in  1662,  0° ;  in 
1700,  9°  40'  W. ;  in  1778,  22°  11'  W. ;  in  1815,  24°  27'  W.  ; 
and  in  1843,  23°  8'  W. 

In  this  country  the  north  end  of  the  needle  was  moving  East- 
ward at  the  earliest  recorded  observations,  and  continued  to  do  so 
till  about  the  year  1810  (variously  recorded  as  from  1793  to 
1819),  when  it  began  to  move  Westward  which  it  has  ever  since 
continued  to  do.  Thus,  in  Boston,  from  1708  to  1807  the  Varia 
tion  changed  from  9°  W.  to  6°  5'  W.,  and  from  1807  to  1840,  it 
changed  from  6°  5'  W.  to  9°  18'  W. 

Valuable  Tables  of  the  Secular  changes  of  the  Variation  in  van 
ous  parts  of  the  United  States  have  been  published  by  Prof.  Loomia 
m  Silliman's  "  American  Journal  of  Science,"  Vol.  34,  July,  1838, 
p.  301 ;  Vol.  39,  Oct.  1840,  p.  42 ;  and  Vol.  43,  Oct.  1842, 
p.  107.  An  abstract  of  the  most  reliable  of  them  is  here  given. 
Troy  and  Schenectady  are  from  other  sources. 

*  If  the  term  "  Declination  of  tlie  Needle"  could  be  restored  to  its  proper  use, 
Jiia  "  Change  of  Variation'  woiftd  be  properly  called  the  "  Variation  of  tlie  De 
clination.'" 


CHAP.  VIII.] 


€lianges  in  the  Variation. 


205 


PLACE. 

LATITUDE. 

LONGITUDE. 

DATES. 

AJ^NUAL 
MOTION. 

[Bui-lington,  Vt. 

44^  27' 

73°  10' 

1811.. .1834 

4'.4 

Chesterfield,  N.  H. 
IDeerfield,  Mass. 

420  53' 

72°  20' 

1820...1836 

6'.4 

42^  34' 

72°  29' 

1811...1837 

5'.7 

iCambridge,  Mass. 

42^  22' 

71°  ^7' 

1810...1840 

3'.4 

New-Haven,  Conn. 

41°  18' 

72°  58' 

1819...1840 

4'.6 

Keeseville,  N.  Y. 

44°  28' 

73°  32' 

1S25...1838 

5'.4    , 

Albany,  N.  Y. 

42°  39' 

73°  45' 

1818...1842 

3'.6 

a 

a 

(( 

1842...1854 

4'.9 

Troy,  N.  Y. 

42°  44' 

73°  40' 

1821...1837 

6'.2 

Schenectady,  N.  Y. 

42°  49' 

730  55' 

1829...1841 

7'.2 

U 

a 

a 

1841.. .1854 

6'!o 

New-York  City. 

40°  43' 

74°  01' 

1824...1837 

3'.7 

Philadelphia. 

39°  57' 

75°  11' 

1813...1837 

3'.6 

Milledge^dlle,  Ga. 

33°    7' 

83°  20' 

1805.. .1835 

1'.7 

Mobile,  Ala. 

30°  40' 

88°  11' 

1809...1835 

2'. 2 

Cleveland,  0. 

41°  30' 

81°  46' 

1825...1838 

4'!5 

Marietta,  0. 

39°  25' 

81°  26' 

1810...1838 

2'.4 

Cincinnati,  0. 

39°    6' 

84°  27' 

1825.. .1840 

2'.0 

Detroit,  Mich. 

42°  24' 

82°  58 

1822...1840 

4'.  3 

Alton,  111. 

38°  52' 

90°  12' 

1835...1840 

3'.0 

From  these  and  other  observations  it  appears  that  at  present  the 
lines  of  equal  variation  are  moving  Westward,  producing  an  annual 
change  of  variation  (increasing  the  Westerly  and  lessening  the 
Easterly)  which  is  different  in  different  parts  of  the  country,  and 
is  about  five  or  six  minutes  in  the  North-Eastern  States,  thi-ee  or 
four  minutes  in  the  Middle  States,  and  two  minutes  in  the  Southern 
States. 


C318)    Determination  of  tlie  change,  by  Interpolation.    To 

determine  the  change  at  any  place  and  for  any  interval  not  found 
in  the  recorded  observations,  an  approximation,  sufficient  for  most 
purposes  of  the  surveyor,  may  be  obtained  by  interpolation  (by  a 
simple  proportion)  between  the  places  given  in  the  Tables,  assum- 
ing the  movements  to  have  been  uniform  between  the  given  dates ; 
and  also  assuming  the  change  at  any  place  not  found  in  the  Tables, 
to  have  been  intermediate  between  those  of  the  lines  of  equal  varia- 
tion, which  pass  through  the  places  of  recorded  observations  on 
each  side  of  it,  and  to  have  been  in  the  ratio  of  its  respective  dis- 


206  COMPASS  SCKVEYIXG.  [part  iii 

tances  from  those  two  lines ;  for  example,  taking  tlieir  aritlimetical 
mean,  if  the  required  place  is  midway  between  them ;  if  it  be  twice 
as  near  one  as  the  other,  dividing  the  sum  of  twice  the  change  of 
the  nearest  Hne,  and  once  the  change  of  the  other,  by  three  ;  and 
80  in  other  cases ;  i.  e.  giving  the  change  at  each  place,  a  "  weight" 
inversely  as  its  distance  from  the  place  at  which  the  change  is  to 
be  found. 

(319)  Determination  of  the  change,  by  old  lines.     When 

the  former  Bearing  of  any  old  line,  such  as  a  farm-fence,  &;c.  is 
recorded,  the  change  in  the  Variation  from  the  date  of  the  origmal 
observation  to  the  present  time  can  be  at  once  found  by  setting  the 
compass  at  one  end  of  the  line  and  sighting  to  the  other.  The 
diiference  of  the  two  Bearings  is  the  required  change. 

If  one  end  of  the  old  line  cannot  be  seen  from  the  other,  as  is 
often  the  case  when  the  line  is  fixed  only  by  a  "corner"  at  each 
end  of  it,  proceed  thus.  Run  a  line  from  one  corner  with  the  old 
.Bearing  and  with  its  distance.  Measure  the  distance  from  the  end 
of  this  line  to  the  other  corner,  to  which  it  will  be  opposite.  Mul- 
tiply this  distance  by  57.3,  and  divide  by  the  length  of  the  line. 
The  quotient  will  be  the  change  of  variation  in  degrees.* 

For  example,  a  line  63  chains  long,  in  1827  had  a  Bearing  of 
North  1*^  East.  In  1847  a  trial  line  was  run  from  one  end  of  the 
former  line  with  the  same  Bearing  and  distance,  and  its  other  end 
was  found  to  be  125  links  to  the  West  of  the  true  corner.     The 

change  of  Variati'm  was  therefore  — -n, '—  =  1°.137  =  1^  8' 

°  bo 

Westerly. 

•  Let  AB  be  the  original  line ;  AC  the  trial  line,  Fig.  208. 

■nd   BC  the  distance    between  their  extremities,    ^t^  ; 

AB  and  AC  may  be  regarded  as  radii  of  a  circle 
and  BC  as  a  chord  of  the  arc  which  subtends  their 
angle.  Assuming  the  chord  and  arc  to  coincide 
(which  tliey  will,  nearly,  for  small  angles)  we 
have  this  proportion  ;  Whole  cir(  iimference  :  arc 
BC  ::  360°  :  BAG  :    or,  2  X  AC  X  3.1416  :  BC 

HP 
•  :  360O  :  BAG,  whence  BAC  =  _    X  57.3  :  or  more  precisely  57.2P578. 


DHAP.  VIII.  j 


Cbanges  in  the  Variation. 


207 


(320)   Effects  of  the  Secular  change.    These  are  exceedingly 
important  in  the  re-survey  of  farms  F'g-  209. 

by  the  Bearings  recorded  in  old  '' 

deeds.  Let  SN  denote  the  direc- 
tion of  the  needle  at  the  time  of 
the  origuial  Survey,  and  S'N'  its 
direction  at  the  time  of  the  re-sur- 
vey, a  number  of  years  later. 
Suppose  the  change  to  have  been 
3°,  the  needle  pointing  so  much 
farther  to  the  west  of  North.  The 
line  SN,  which  before  was  due 
North  and  South  by  the  needle 
will  now  bear  N.  3°  E.  and  S.  3°  W ;  the  line  AB,  which  before 
was  N.  40°  E.  will  now  bear  N.  43°  E ;  the  line  DF  which  before 
was  N.  40°  W.  will  now  bear  N.  37°  W;  and  the  line  WE,  which 


87°  W.  Any  line  is  similarly  changed.  The  proof  of  this  is  appar 
rent  on  inspecting  the  figure. 

Suppose  now  that  a  sui'veyor,  ignorant  or  neglectful  of  this 
change,  should  attempt  to  run  out  a  F'=-  -^^• 

farm  by  the  old  Bearuigs  of  the 
deed,  none  of  the  old  fences  or  cor- 
ners ^emaiuing.  The  full  lines  in 
the  figure  represent  the  original  / 
bounds  of  the  farm,  and  the  dotted  \ 
lines  those  of  the  new  piece  of  land 
which,  starting  from  A,  he  would 
imwittingly  run  out.  It  would  be  of 
the  same  size  and  the  same  shape  aa 
the  true  one,  but  it  would  be  in  the 
wrong  place.  None  of  its  lines 
would  agree  with  the  true  ones,  and 
in  some  places  it  would  encroach  on 
one  neighbor,  and  in  other  places 

would  leave  a  gore  which  belongs  to  It,  between  itself  and  another 
neighbor.  Yet  this  is  often  done,  and  is  the  source  of  a  gi-eat  part 
of  the  litigation  among  farmers  respecting  their  "  lines." 


208  COMPASS  SURVEYING.  [part  hi. 

(321)  To  run  out  old  lines,  To  succeed  in  retracing  old 
lines,  proper  allowance  must  be  made  for  the  change  in  the  varia- 
tion since  the  date  of  the  original  survey.  That  date  must  first 
be  accurately  ascertained ;  for  the  survey  may  be  much  older  than 
the  deed,  into  which  its  bearings  may  have  been  copied  from  an 
older  one.  The  amount  and  direction  of  the  change  is  then  to  be 
ascertained  by  the  methods  of  Arts.  (318)  or  (319).  The  bear- 
ings may  then  be  corrected  by  the  following  Rules. 

When  the  North  end  of  the  needle  has  been  moving  Westerly, 
(as  it  has  for  about  forty  years),  the  present  Bearings  will  be  the 
sums  of  the  change  and  the  old  Bearings  which  were  North-East- 
erly or  South- Westerly,  and  the  differences  of  the  change  and 
the  old  Bearings  which  were  North- Westerly  or  South-Easterly. 

If  the  change  have  been  Easterly,  reverse  the  preceding  rules, 
subtracting  where  it  is  directed  to  add,  and  adding  where  it  is 
directed  to  subtract. 

Run  out  the  lines  with  the  Bearings  thus  corrected. 

It  will  be  noticed  that  the  process  is  precisely  the  reverse  of 
that  in  Art.  (311)  The  rules  there  given  in  more  detail,  may 
therefore  be  used;  Rule  1,  "  when  the  Variation  is  West,"  being 
employed  when  the  change  has  been  a  movement  of  the  N.  end 
of  the  needle  to  the  East ;  and  Rule  2,  "  when  the  Variation  is 
East,"  being  employed  when  the  N.  end  of  the  needle  has  been 
moving  to  the  West. 

If  the  compass  has  a  Vernier,  it  can  be  set  for  the  change,  once 
for  all,  precisely  as  directed  in  Art.  (312),  and  then  the  courses 
can  be  run  out  as  given  in  the  deed,  the  correction  being  made  by 
the  instrument. 


(322)  Example.  The  following  is  a  remarkable  case  which 
recently  came  before  the  Supreme  Court  of  New-York.  The 
North  line  of  a  large  Estate  was  fixed  by  a  royal  grant,  dated  in 
1704,  as  a  due  East  and  West  line.  It  was  run  out  in  1715, 
by  a  surveyor,  whom  we  will  call  Mr.  A.  It  was  again  surveyed 
in  1765,  by  Mr.  B.  who  ran  a  course  N.  87°  30'  E.  It  was  run 
out  for  a  third  time  in  1789,  by  Mr.  C.  who  adopted  the  course 
N.  86°  18'  E.     In  1845  it  was  surveyed  for  the  fourth  time  by 


CHAP.  VIII.]  Changes  in  the  Variation.  209 

Mr.  D.  with  a  course  of  N.  88"  30'  E.  He  found  old  "  corners," 
and  "  blazes"  of  a  former  survey,  on  his  line.  They  are  also  found 
on  another  line.  South  of  his.  AVhich  of  the  preceding  courses 
were  correct,  and  where  does  the  true  line  lie  ? 

The  question  was  investigated  as  follows.  There  wore  no  old 
records  of  variation  at  the  precise  locahty,  but  it  lies  between  the 
lines  of  equal  variation  which  pass  through  New-York  and  Boston, 
its  distance  from  the  Boston  Une  being  about  twice  its  distance  from 
the  New- York  hue.  The  records  of  those  two  cities  (referred  to 
in  Art.  (317))  could  therefore  be  used  in  the  manner  explained 
in  Art.  (318).  For  the  later  dates,  observations  at  New-Haven 
could  serve  as  a  check.  Combining  all  these,  the  author  inferred 
the  variation  at  the  desired  place  to  have  been  as  follows : 
In  1715,  Variation  8°  02'  West. 

In  1765,         "        50  32'      "  Decrease  since  1715,  2°  30'. 

In  1789,         "        50  05'      "  Decrease  since  1765,  0^  27'. 

In  1815,         "        70  23'      "  Increase  since  1789,  2°  18'. 

We  are  now  prepared  to  examine  the  correctness  of  the  allowances 
made  by  the  old  surveyors. 

The  course  run  by  Mr.  B.  in  1765,  N.  87°  30'  E.  made  an 
allowance  of  2°  30'  as  the  decrease  of  variation,  agreeing  precisely 
with  our  calculation.  The  course  of  Mr.  C.  in  1789,  N.  86°  18' 
E.,  allowed  a  change  of  1°  12',  which  was  wrong  by  our  calcula- 
tion, which  gives  only  about  27',  and  was  deduced  from  three  dif- 
ferent records.  Mr.  D.  in  1815,  ran  a  course  of  N.  88°  30'  E, 
calUng  the  increase  of  variation  since  1789,  2^  12'.  Our  estimate 
was  2°  18',  the  difference  being  comparatively  small.  Our  con- 
clusion then  is  this :  the  second  surveyor  retraced  correctly  the  line 
of  the  first :  the  third  surveyor  ran  out  a  neiv  and  incorrect  line :  and 
the  fourth  surveyor  correctly  retraced  the  line  of  the  third,  and  found 
his  marks,  but  this  line  was  wrong  originally  and  therefore  wrong 
now.  All  the  surveyors  ran  their  lines  on  the  supposition  that  the 
original  "  due  East  and  West  line  "  meant  East  and  West  as  the 
needle  pointed  at  the  time  of  the  original  survey. 

The  preponderance  of  the  testimony  as  to  old  land  marJis  agreed 
vnth  the  resvdts  of  the  above  reasoning,  and  the  decision  of  the 
Court  was  in  accordance  therewith. 

11 


210 


COMPASS  SURVEYING. 

Fig.  211 


In  the  above  figure  the  horizontal  and  vertical  Hnes  represent 
true  East  and  North  lines  ;  and  the  two  upper  lines  ruiming  from  left 
to  right  represent  the  two  lines  set  out  by  the  surveyors  and  in  the 
years,  there  named. 

(323)  Remedy  for  the  evils  of  the  Secular  ehan^e.      The 

only  complete  remedy  for  the  disputes,  and  the  uncertainty  of 
bounds,  resulting  from  the  continued  change  in  the  variation,  is 
this.  Let  a  Meridian,  i.  e.  a  true  North  and  South  line,  be  estab« 
lished  in  every  town  or  county,  by  the  authority  of  the  State  ; 
monuments,  such  as  stones  set  deep  in  the  ground,  being  placed  at 
each  end  of  it.  Let  every  surveyor  be  obliged  by  law  to  test  his 
compass  by  this  line,  at  least  once  in  each  year.  This  he  could 
do  as  easily  as  in  taking  the  Bearing  of  a  fence,  by  setting  his 
instrument  on  one  monument,  and  sighting  to  a  staff  held  on  the  other. 
Let  the  variation  thus  ascertained  be  inserted  in  the  notes  of  the 
survey  and  recorded  in  the  deed.  Another  surveyor,  years  or 
centuries  afterwards,  could  test  his  compass  by  taking  the  Bearing 
of  the  same  monuments,  and  the  difference  between  this  and  the 
former  Bearing  would  be  the  change  of  variation.  He  could  thus 
determine  with  entire  certainty  the  proper  allowance  to  be  made 
(as  in  Art.  (321))  in  order  to  retrace  the  original  hne,  no  matter 
how  much,  or  how  irregularly,  the  variation  may  have  changed,  or 
how  badly  adjusted  was  the  compass  of  the  original  survey.  Any 
permanent  line  employed  in  the  same  manner  as  the  meridian  line, 
would  answer  the  same  purpose,  though  less  conveniently,  and  everj 
surveyor  should  have  such  a  line  at  least,  for  his  own  use.* 

"  riiis  remedy  seems  to  have  been  first  suggested  hv  Ritteiibouse.  It  has  since 
been  recommended  by  T.  Sopwith,  in  ISOS*;  by  E.  F.  Johnson,  in  1831,  and  by 
W,  Roberts,  of  Troy,  in  1839.  The  errors  of  re-surveys,  in  which  the  change 
is  neglected,  were  noticed  in  the  "  Philosoi^hical  Transactions,"  as  long  ago  as  1679 


PAET  IV. 

TRANSIT  AND  THEODOLITE  SURVEYING: 

By  the  Third  Method. 


CHAPTER  I. 


TOE  L\STRIIMENTS. 

(324)  The  Transit  and  The  Theodolite  (figures  of  which  are 
given  on  the  next  two  pages)  are  G-oniometers,  or  Angle-Measurers. 
Each  consists,  essentially,  of  a  circular  plate  of  metal,  supported  in 
Buch  a  manner  as  to  be  horizontal,  and  divided  on  its  outer  circum- 
ference into  degrees,  and  parts  of  degrees,  Through  the  centre  of 
this  plate  passes  an  upright  axis,  and  on  it  is  fixed  a  second  circu- 
.ar  plate,  which  nearly  touches  the  first  plate,  and  can  turn  freely 
around  to  the  right  and  to  the  left.  Tliis  second  plate  carries  a 
Telescope,  which  rests  on  upright  standards  firmly  fixed  to  the 
plate,  and  which  can  be  pomted  upwards  and  downwards.  By 
the  combination  of  this  motion  and  that  of  the  second  plate  around 
.its  axis,  the  Telescope  can  be  directed  to  any  object.  The  second 
plate  has  some  mark  on  its  edge,  such  as  an  arrow-head,  which 
serves  as  a  pointer  or  index  for  the  divided  circle,  like  the  hand  of 
a  clock.  When  the  Telescope  is  directed  to  one  object,  and  then 
turned  to  the  right  or  to  the  left,  to  some  other  object,  this  index, 
which  moves  with  it  and  passes  around  the  divided  edge  of  the 
other  plate,  pouits  out  the  arc  passed  over  by  this  change  of  direc- 
tion, and  thus  measures  the  angle  made  by  the  fines  imagined  to 
pass  from  the  centre  of  the  instrument  to  the  two  objects. 


212 


TRAXSIT  Ai\D  THEODOLITE  SURVEYIXG.    [part  i> 

THE  TRANSIT. 


Fig  212 


CHAP.  I.J  The  Instruments. 

THE  THEODOLITE. 

FiR.  213 


213 


214  TRANSIT  AND  THEODOLITE  SURVEYING,    [part  i^ 

(325)  Distinction.  The  preceding  description  applies  to  both 
the  Transit  and  the  Theodolite.  But  an  essential  diflference 
between  them  is,  that  in  the  Transit  the  Telescope  can  turn  com 
pletely  over,  so  as  to  look  both  forward  and  backward,  while  iu 
the  Theodolite  it  cannot  do  so.     Hence  the  name  of  the  Transit.* 

This  capabUitj  of  reversal  enables  a  straight  line  to  be  prolonged 
from  one  end  of  it,  or  to  be  ranged  out  in  both  directions  from  any 
one  point.  The  Telescope  of  the  Theodolite  can  indeed  be  taken 
out  of  the  Y  shaped  supports  in  which  it  rests,  and  be  replaced 
end  for  end,  but  this  operation  is  an  imperfect  substitute  for  the 
revolution  of  the  Telescope  of  the  Transit.  So  also  is  the  turning 
half  way  around  of  the  upper  plate  which  carries  the  Telescope. 

Tlie  Theodolite  has  a  level  attached  to  its  Telescope,  and  a  vertical 
circle  for  measuring  vertical  angles.  The  Transit  does  not  usually 
have  these,  though  they  are  sometimes  added  to  it.  The  instru- 
ment may  then  be  named  a  Transit-Theodolite.  It  then  corre- 
sponds to  the  altitude  and  azimuth  instrument  of  Astronomy.  As 
the  greater  part  of  the  points  to  be  explained  are  common  to  both 
the  Transit  and  the  Theodolite,  the  descriptions  to  be  given  may 
be  regarded  as  applicable  to  either  of  the  instruments,  except  when 
the  contrary  is  expressly  stated,  and  some  point  peculiar  to  either 
is  noticed. 

(326)  The  great  value  of  these  instruments,  and  the  accuracy 

of  their  measurements  of  angles  are  due  chiefly  to  two  things ;  to 
the  Telescope,  by  which  great  precision  in  sighting  to  a  point  is 
obtained  ;  and  to  the  Vernier  Scale,  which  enables  minute  portions 
of  any  arc  to  be  read  with  ease  and  correctness.  The  former 
assists  the  eye  in  directing  the  line  of  sight,  and  the  latter  aids  it 
in  reading  off  the  results.  Arrangements  for  giving  slow  and 
steady  motion  to  the  movable  parts  of  the  instruments  add  to  the 
value  of  the  above.  A  contrivance  for  Repeating  the  observation 
of  angles  still  farther  lessens  the  unavoidable  inaccuracies  of 
these  observations. 

•  It  is  sometimes  called  the  "  Engineers'  Transit,"  or  "  Railroad  Transit,"  to 
distinguish  it  from  the  Astronomical  Transit-instrument.  In  this  country  it  has 
•almost  entirely  supplanted  the  Theodolite. 


CHAP,  i]  The  Iniitrumeiits.  215 

The  inaccurate  division  of  the  limb  of  the  instrument  is  alsc 
averaged  and  thus  diminished  bj  the  last  arrangement.  Its  want 
of  true  "  centring,"  is  remedied  by  reading  off  on  opposite  sides 
of  the  circle. 

Imperfections  m  the  parallelism  and  perpendicularity  of  the  parts 
of  the  instrument  in  wli  ".h  those  qualities  are  required,  are  cor- 
rected by  various  "  adjustments,"  made  by  the  various  screws 
whose  heads  appear  in  the  engravings. 

The  arrangements  for  attaining  all  these  objects  render  necessary 
the  numerous  parts  and  apparent  complication  of  the  instrument. 
But  this  complication  disappears  when  each  part  is  examined  in 
turn,  and  its  uses  and  relations  to  the  rest  are  distinctly  indicated. 
This  we  now  propose  to  do,  after  explaining  the  engravings. 

(32?)  In  the  figures  of  the  instruments,  given  on  pages  212  and 
213,  the  same  letters  refer  to  both  figures,  so  far  as  the  parts  are 
common  to  both.*  L  is  the  limb  or  divided  circle.  V  is  the 
index,  or  "Vernier,"  which  moves  around  it.  In  the  Transit,  only 
a  small  portion  of  the  divided  hmb  is  seen,  the  upper  circle  (which 
in  it  is  the  movable  one)  covering  it  completely,  so  that  only  a 
short  piece  of  the  arc  is  visible  through  an  opening  in  the  upper 
plate.  S,  S,  are  standards,  fastened  to  the  upper  plate  and  sup- 
porting the  telescope,  EO.  G  is  a  compass-box,  also  fastened  to 
the  upper  plate,  c  is  a  clamp-screw,  which  presses  together  the  two 
plates,  and  prevents  one  from  moving  over  the  other,  tha,  tangent- 
screw,  or  slow-motion  screw,  which  gives  a  slow  and  gentle  motion 
to  one  plate  over  the  other.  C  is  a  clamp-screw  which  fastens  the 
lower  plate  to  the  body  of  the  instrument,  and  thus  prevents  it  from 
moving  on  its  own  axis.  T  is  the  tangent-screw  to  give  this  part  a 
slow-motion.  P  and  P'  are  parallel  plates  through  which  pass  foui 
screws,  Q,  Q,  Q,  Q,    by  which  the  circular  plate  L  is  made  level. 

*  Tlie  arraugements  of  these  instruments  are  differently  made  by  almost  every 
maker;  bin  any  form  of  them  being  thoroughly  understood,  any  new  one  will 
cause  no  difficulty.  The  figure  of  the  Transit  was  drawn  from  one  made  by  W. 
&  L.  E.  Gurley,  of  Troy,  N.  Y.  to  the  latter  of  whom  the  Author  is  indebted  for 
Bome  valuable  information  respecting  the  details  of  the  instrument.  The  Theodo 
lit3  is  of  the  favorite  English  form. 


216        TRA]\SIT  MD  THEODOLITE  SURVEYIXG.       [pakt  iv 

as  determined  bj  the  bubbles  iu  the  small  spirit  levels,  B,  B,  of 
which  there  are  two  at  right  angles  to  each  other. 

In  the  figure  of  the  Theodolite,  the  large  level  b,  and  the  semi* 
circle  NN  are  for  the  purposes  of  Levelhng,  and  of  measurino 
Vertical  angles.     Thej  will  therefore  not  be  described  in  this  place. 


(328)  As  the  value  of  either  of  these  instruments  dependa 
gi'eatlj  on  the  accurate  fitting  and  bearings  of  the  two  concentric 
vertical  axes,  and  as  their  connection  ought  to  be  thoroughly  under- 
stood, a  vertical  section  through  the  body  of  the  instrument  is 
given  in  Fig.  214,  to  half  the  real  ^ize.     The  tapering  spindle  or 

Fig.  2]  4. 


Tr^ 


mverted  frustum  of  a  cone,  marked  AA,  supports  the  upper  plate 
BB,  which  carries  the  index,  or  Verniers,  V,  V,  and  the  Telescope. 
The  whole  bearing  of  this  plate  is  at  C,  C,  on  the  top  of  the  hollow 
inverted  cone  EE,  in  which  the  spindle  turns  freely,  but  steadily. 
This  interior  position  of  the  bearings  preserves  them  from  dust 
and  injury.  This  hollow  cone  carries  the  lower  or  graduated  plate, 
and  it  can  itself  turn  around  on  the  bearings  D,  D,  carrying  with  it 
the  lower  circle,  and  also  the  upper  one  and  all  above  it. 

The  Vernier  scales  V,  V,  are  attached  to  the  upper  plate,  but 
lie  in  the  same  plane  as  the  divisions  L,  L,  of  the  lower  plate,  (so 
khat  the  two  can  be  viewed  together,  without  parallax,)  and  are 


tHAP.  I.] 


The  Instruments. 


217 


covered  with  glass,  to  exclude  dust  and  moisture.  In 
the  figure  the  hatcliings  are  drawn  in  different  directions 
on  the  parts  which  move  Avith  the  Vernier,  and  on 
those  which  move  only  with  the  limb. 


(329)  The  Telescope.  This  is  a  combination  of 
lenses,  placed  in  a  tube,  and  so  arranged,  in  accordance 
with  the  laws  of  optical  science,  that  an  image  of  any 
object  to  which  the  Telescope  may  be  directed,  is  formed 
withm  the  tube,  (by  the  rays  of  light  coming  from 
the  object  and  bent  in  passing  through  the  object-glass) 
and  there  magnified  by  an  Eye-glass,  or  Eye-piece,  com- 
posed of  several  lenses.  The  arrangement  of  these  lenses 
are  very  various.  Those  two  combinations  which  are  pre- 
ferred for  survejdng  instruments,  will  be  here  explained. 

Fig.  215  represents  a  Telescope  which  inverts  objects. 
Any  object  is  rendered  visible  by  every  point  of  it  send- 
ing forth  rays  of  Ught  in  every  direction.  In  this  figure, 
the  highest  and  lowest  points  of  the  object,  which  here  is 
an  arrow.  A,  are  alone  considered.  Those  of  the  rays 
proceeding  from  them,  which  meet  the  object-glass,  0, 
form  a  cone.  The  centre  fine  of  each  cone,  and  its  ex- 
treme upper  and  lower  lines  are  alone  shown  in  the 
figure.  It  wiU  be  seen  that  these  rays,  after  passing 
through  the  object-glass,  are  refracted,  or  bent,  by  it, 
so  as  to  cross  one  another,  and  thus  to  form  at  B  an 
mverted  image  of  the  object.  This  would  be  rendered 
visible,  if  a  piece  of  gi-ound  glass,  or  other  semi-transpa- 
rent substance,  was  placed  at  the  point  B,  which  is  called 
the  focus  of  the  object-glass.  The  rays  which  form  this 
image  continue  onward  and  pass  through  the  two  lenses 
C  and  D,  which  act  like  one  magnifying  glass,  so  that 
the  rays,  after  being  refracted  by  them,  enter  the  eye 
at  such  angles  as  t'>  form  there  a  magnified  and  invert- 
ed image  of  the  object.  This  combination  of  the  two 
plano-convex  lenses,  C  and  D,  is  known  as  "Kamsden's 
Ey>piece." 


o  fi' 


Z&jV 


?18        TRAASIT  AXD  THEODOLITE  SIRVEYL\G.      [pak*  it 


This  Telescope,  inverting  objects,  shows  them  upside 
down,  and  the  right  side  on  the  left.  They  can  be 
shown  erect  bj  adding  one  or  two  more  lenses  as  in  the 
marginal  figure.  But  as  these  lenses  absorb  light  and  les- 
sen the  distinctness  of  vision,  the  former  arrangement  is 
preferable  for  the  glasses  of  a  Transit  or  a  Theodolite. 
A  little  practice  makes  it  equally  convenient  for  the 
observer,  who  soon  becomes  accustomed  to  seeing  his 
flagmen  standing  on  their  heads,  and  soon  learns  to 
motion  them  to  the  right  when  he  wishes  them  to  go  to 
the  left,  and  vice  versa. 

Figure  216  represents  a  Telescope  which  shows  ob- 
jects erect.  Its  eye-piece  has  four  lenses.  The  eye- 
piece of  the  common  terrestrial  Telescope,  or  spy-glass, 
has  three.  Many  other  combinations  may  be  used,  all 
intended  to  show  the  object  achromatically,  or  free  from 
false  coloring,  but  the  one  here  shown  is  that  most  gene- 
rally preferred  at  the  present  day.  It  will  be  seen  that 
an  inverted  image  of  the  object  A,  is  formed  at  B,  as 
before,  but  that  the  rays  continuing  onward  are  so 
refracted  in  passing  through  the  lens  C  as  to  again 
cross,  and  thus,  after  farther  refraction  by  the  lenses  D 
and  E,  to  form,  at  F,  an  erect  image,  which  is  magni- 
fied by  the  lens  G. 

In  both  these  figures,  the  limits  of  the  page  render 
it  necessary  to  draw  the  angles  of  the  rays  very  much 
out  of  proportion. 

(330)  Cross-hairs.  Since  a  considerable  field  of 
view  is  seen  in  looking  through  the  Telescope,  it  is 
necessary  to  pro\dde  means  for  directing  the  line  of  sight 
to  the  precise  point  which  is  to  be  observed.  This 
30uld  be  effected  by  placing  a  very  fine  point,  such  as 
that  of  a  needle,  within  the  Telescope,  at  some  place 
where  it  could  be  distinctly  seen.  In  practice  this  fine 
point  is  obtained  by  the  intersection  of  two  very  fine 
lines,  placed  in  the  common /oc?«c  of  the  object-glass  and 


nflAP.  I.] 


The  Instruments. 


219 


Fiff.  217. 


ot  the  eye-piece.  These  lines  are  called  the  cross-hairs,  or  crosS' 
wires.  Their  intersection  can  be  seen  through  the  eje-piece,  at 
the  same  time,  and  apparently  at  the  same  place,  as  the  image  of  the 
distant  object.  The  magnifying  powers  of  the  eye-piece  will  then  de- 
tect the  slightest  deviation  from  perfect  coincidence.  "  This  apphca- 
tion  of  the  Telescope  may  be  considered  as  completely  annihilat- 
mg  that  part  of  the  error  of  observation  which  might  otherwise 
arise  from  an  erroneous  estimation  of  the  direction  in  Avhich  an 
object  lies  from  the  observer's  eye,  or  from  the  centre  of  the  instru- 
ment. It  is,  in  fact,  the  grand  source  of  all  the  precision  of  modern 
Astronomy,  without  which  all  other  refinements  in  instrumental  work 
manship  would  be  thrown  away."  "WTiat  Sir  John  Herschel  here 
says  of  its  utility  to  Astronomy,  is  equally  applicable  to  Surveying. 

The  imaginary  line  which  passes  through  the  intersection  of  th*^ 
cross-hairs  and  the  optical  centre  of  the  object-glass,  is  called  the 
line  of  collimation  of  the  Telescope.* 

The  cross-hairs  are  attached  to  a  ring,  or  short  thick  tubft  of 
brass,  placed  within  the  Tele- 
scope tube,  through  holes  in 
which  pass  loosely  four  screws, 
(their  heads  being  seen  at  a, 
a,  a,  in  Figs.  212  and  213), 
whose  threads  enter  and  take 
hold  of  the  ring,  behind  or  in 
front  of  the  cross-hairs,  as 
shown  (in  front  view  and  in 
section)  in  the  two  figures  in 
me  margin.     Their  movements  will  be  explained  in  Chapter  III. 

Usually,  one  cross-hair  is  horizontal,  and  tho 
other  vertical,  as  in  Fig.  217,  but  sometimes  they 
are  arranged  as  in  Fig.  218,  which  is  thought  to 
enable  the  object  to  be  bisected  with  more  preci- 
sion.    A  horizontal  hair  is  sometimes  added. 

The  cross-hairs  are  best  made  of  platmum  wire, 
drawn  out  very  fine  by  bemg  previously  enclosed 

•  From  the  Latin  word  Collimo,  or  CoUineo,  meaning  to  direct  one  thing  to 
wards  another  in  a  straight  line,  or  to  aim  at.  The  line  of  aim  would  express  ilia 
meaning. 


218. 


WWV 


220        TRIXSIT  A\D  THEODOLITE  SFRVEmO.      L^^art  it. 

in  a  larger  wire  of  silver,  and  the  silver  then  removed  by  nitric 
acid.  Silk  threads  from  a  cocoon  are  sometimes  used.  Spiders' 
threads  are,  however,  the  most  usual.  If  a  cross-hair  is  broken, 
the  ring  must  be  taken  out  by  removing  two  opposite  screws,  and 
inserting  a  \\'ire  with  a  screw  cut  on  its  end,  or  a  stick  of  suitable 
size,  into  one  of  the  holes  thus  left  open  in  the  ring,  it  being  turned 
sideways  for  that  purpose,  and  then  removing  the  other  screws . 
The  spider's  threads  are  then  stretched  across  the  notches  seen  in 
the  end  of  the  ring,  and  are  fastened  by  gum,  or  varnish,  or  bees- 
wax. The  operation  is  a  very  delicate  one.  The  following  plan  has 
been  employed.  A  piece  of  wire  is  bent,  as  in  the  figure,  so  as  to 
leave  an  opening  a  little  wider  than  the  V'?.  219. 

ring  of  the  cross-hairs.  A  cobweb  is  cho- 
sen, at  the  end  of  which  a  spider  is  hang- 
ing, and  it  is  wound  around  the  bent  wire,  ^ 
as  m  the  figure,  the  weight  of  the  insect  ~^ 
keeping  it  tight  and  stretching  it  ready  for  use,  each  part  being 
made  fast  by  gum,  &c.  When  a  cross-hair  is  wanted,  one  of 
these  is  laid  across  the  ring  and  there  attached.  Another  method 
is  to  draw  the  thread  out  of  the  spider,  persuading  him  to  spm,  if 
he  sulks,  by  tossing  him  from  hand  to  hand.  A  stock  of  such 
threads  must  be  obtained  in  warm  weather  for  the  winter's  wants. 
A  piece  of  thin  glass,  with  a  horizontal  and  a  vertical  line  etched 
on  it,  may  be  made  a  substitute. 

(331)  Instramental  Parallax.  This  is  an  apparent  movemem 
of  the  cross-hairs  about  the  object  to  which  the  line  of  sight  is 
directed,  taking  place  on  any  shght  movement  of  the  eye  of  the 
observer.  It  is  caused  by  the  image  and  the  cross-hairs  not  being 
precisely  in  the  common  focus,  or  point  of  distinct  \4sion  of  the 
eye-piece  and  the  object  glass.  To  correct  it,  move  the  eye-piece 
out  or  in  tiU  the  cross-hairs  are  seen  clearly  and  sharply  defined 
against  any  white  object.  Then  move  the  object  glass  in  or  out 
till  the  object  is  also  distmctly  seen.  The  cross-hairs  will  then 
seem  to  be  fixed  to  the  object,  and  no  movement  of  the  eye  will 
cause  them  to  appear  to  change  their  place. 


CHAP  ;.]  The  Instruments.  22i 

(332)  The  milled-headed  screw  seen  at  M,  passing  into  the  tele 

scope  has  a  pinion  at  its  other  end  entering  a '^ 'p-.!^-"-  _1 

toothed  rack,  and  is  used  to  move  the  object  glass,    ''^-'wvaaAAA'x.-*.^/ 
0,  out  and  in,  according  as  the  object  looked  at  is  %a^ 

nearer  or  farther  than  the  one  last  observed.     Short  distancca 
require  a  long  tube :  long  distances  a  short  tube. 

The  eye-piece,  E,  is  usually  moved  in  and  out  by  hand,  but  a 
similar  arrangement  to  the  preceding  is  a  great  improvement.  This 
movement  is  necessary  in  order  to  obtain  a  distinct  \'ie\v  of  the 
cross-hairs.  Short-sighted  persons  require  the  eye-picce  to  be 
pushed  farther  in  than  persons  of  ordinary  sight,  and  old  or  long- 
sighted persons  to  have  it  drawn  further  out. 

(333)  Supports,  The  Telescope  of  the  Transit  is  supported 
by  a  hollow  axis  at  right  angles  to  it,  which  itself  rests  at  each  end, 
on  tAvo  upright  pieces,  or  standards,  spreading  at  their  bases  so  as 
to  increase  their  stability.  In  the  Theodolite,  the  telescope  rests 
at  each  end  in  forked  supports,  called  ys,  from  their  shape. 
These  Ys  are  themselves  supported  by  a  cross-bar,  which  is  car- 
ried by  an  axis  at  right  angles  to  it  and  to  the  telescope.  This' 
axis  rests  on  standards  similar  to  those  of  the  Transit.  The  Tele- 
scope of  the  Theodolite  can  be  taken  out  of  the  Ys,  and  turned 
"  end  for  end."  This  is  not  usual  in  the  Transit.  Either  of  the 
above  arrangements  enables  the  Telescope  to  be  raised  or  depressed 
BO  as  to  suit  the  height  of  the  object  to  which  it  is  directed.  A 
telescope  so  disposed  is  called  a  "  plunging  telescope." 

In  some  instruments  there  is  an  arrangement  for  raising  or  low- 
ering one  end  of  the  axis.  This  is  sometimes  requu-ed  for  reasons 
to  be  given  in  connection  with  "  Adjustments." 

(334)  The  Indexes.  Thesupports,  or  standards,  of  the  telesco^je 
just  described  are  attached  to  the  upper,  or  index-carrj-ing  circle.* 
This,  as  has  been  stated,  can  turn  freely  on  the  lower  or  graduated 
circle,  by  means  cf  its  conical  axis  moving  in  the  hollow  conical 
axis  of  the  latter  circle.     This  upper  circle  carries  the  index,  V, 

•  In  some  insiruineiUs  this  circle  is  the  under  one.  In  our  figures  it  is  the  uppei 
one.  and  we  will  therefore  always  speak  of  it  as  such. 


222        TRAIVSIT  AXD  THEODOLITE  SURVEYING.        [part  iv 


which  is  an  arrow-head  or  other  mark  on  its  edge,  or  the  zero-point  of 
a  Vernier  scale.  There  are  usually  two  of  these,  situated  exactly 
opposite  to  each  other,  or  at  the  extremities  of  a  diameter  of  the 
upper  circle,  so  that  the  readings  on  the  graduated  circle  pointed  out 
by  them  differ,  if  both  are  correct,  exactly  180°.  The  object  of  this 
arrangement  is  to  correct  any  error  of  eccentricity,  arising  from  the 
centre  of  the  axis  which  carries  the  upper  circle,  (and  with  which  it 
and  its  index  pointers  turn),  not  being  precisely  in  the  centre  of  the 
graduated  circle.  In  the  figure,  let  C 
be  the  true  centre  of  the  graduated  cir- 
cle, but  C  the  centre  on  which  the  plate 
carrying  the  indexes  turns.  Let  AC'B 
represent  the  direction  of  a  sight  taken 
to  one  object,  and  D'C'E'  the  direction 
when  turned  to  a  second  object.  The 
angle  subtended  by  the  t^vt)  objects  at 
the  centre  of  the  instrument  is  requir- 
ed. Let  DE  be  a  line  passing  through  C,  and  parallel  to  D'E'. 
The  angle  ACD  equals  the  required  angle,  which  is  therefore  truly 
measured  by  the  arc  AD  or  BE.  But  if  the  arc  shown  by  the 
index  is  read,  it  will  be  AD'  on  one  side,  and  BE'  on  the  other ; 
the  first  being  too  small  by  the  arc  DD'  and  the  other  too  large  by- 
the  equal  arc  EE'.  If  however  the  half-sum  of  the  two  arcs  AD' 
and  BE'  be  taken,  it  will  equal  the  true  arc,  and  therefore  correctly 
measure  the  angle.  Thus  if  AD'  was  19°,  and  BE'  21°,  their 
half  sum,  20°,  would  be  the  correct  angle. 

Three  indexes,  120°  apart,  are  sometimes  used.  They  have  th? 
advantage  of  averaging  the  unavoidable  inaccuracies  and  inequali 
ties  of  graduation  on  different  parts  of  the  limb,  and  thus  diminish 
ing  their  effect  on  the  resulting  angle. 

Four  were  used  on  the  large  Theodolite  of 
the  English  Ordnance  Survey,  two,  A  and  B, 
opposite  to  each  other,  and  two,  C  and  D,  120° 
from  A  and  from  each  other.  The  half-sum  or 
arithmetical  mean,  of  A  and  B  was  taken,  then 
the  mean  of  A,  C,  and  D,  and  then  the  mean 
of  these  two  means.     But  this  was  wrong,  for 


cuAF.  1  ]  The  lustruments.  223 

it  gave  too  great  value  to  the  reading  of  A,  and  also  to  B,  though 
in  a  less  degree  ;  since  the  share  of  each  Vernier  in  the  final  mean 
was  as  follows :  A  =  5,  B  =  8,  C  ^  2,  D  =  2.     This  results  from 


the  expression  for  that  mean,  =  ^  I 
(5  A  +  3  B  +  2  C  +  2  D). 


A  +  B       A  +-C  +  D^ 


T2 


(335)  The  graduated  circle.  This  is  divided  into  three  hun- 
dred and  sixty  equal  parts,  or  Degrees,  and  each  of  these  is  sub- 
divided into  two  or  three  parts  or  more,  according  to  the  size  of  the 
instrument.  In  the  first  case,  the  smallest  division  on  the  circle  will 
of  course  be  30';  in  the  second  case  20'.  More  precise  reading,  to 
single  minutes  or  even  less,  is  eflfected  by  means  of  the  Vernier  of 
the  index,  all  the  varieties  of  which  will  be  fully  explained  in  the 
next  chapter.  The  numbers  run  from  0--'  around  to  360^,  which 
Qumber  is  necessarily  at  the  same  point  as  the  0,  or  zero-point.* 
Each  tenth  degree  is  usually  numbered,  each  fifth  degree  is  distin- 
guished by  a  longer  line  of  division,  and  each  degree-division  fine 
is  longer  than  those  of  the  sub-divisions.  A  magnifying  glass  is 
ueeded  for  reading  the  divisions  with  ease.  In  the  Theodohte 
engraving  this  is  shown  at  m.  It  should  be  attached  to  each 
Vernier. 

(336)  Movements.  When  the  fine  of  sight  of  the  telescope  is 
directed  to  a  distant  well-defined  point,  the  unaided  hand  of  the 
observer  cannot  move  it  with  sufficient  delicacy  and  precision  to 
make  the  intei'section  of  the  cross  hairs  exactly  cover  or  "bisect" 
that  point.  To  efiect  this,  a  clamp,  and  a  Tangent,  or  slow-motion, 
screw  are  required.  This  arrangement,  as  appfied  to  the  move- 
ment of  the  upper,  or  Vernier  plate,  consists  of  a  short  piece  of 
brass,  D,  which  is  attached  to  the  Vernier  plate,  and  through 
which  passes  a  long  and  fine-threaded  "  Tangent-screw,"  t.  The 
other  end  of  this  screw  enters  into  and  carries  the  clamp.  This 
consists  of  two  pieces  of  brass,  which,  by  turning  the  clamp-screw 
tf,  which  passes  through  them  on  the  outside,  can  be  made  to  take 

•  In  some  instruments  there  is  another  concentric  ciicle  on  wliitli  tiie  degiees 
are  also  tiuiubcred  from  0°  to  90o  as  on  the  compass  circle. 


224        TRANSIT  A\D  THEODOLITE  SlRVEYIiVG.       [part  iv 

hold  of  and  pinch  tightly  the  edge  of  the  louver  circle,  which  hes 
hetween  them  on  the  inside.  The  upper  circle  is  now  prevented 
from  moving  on  the  lower  one  ;  for,  the  tangent-screw,  passing 
through  hollow  screws  in  both  the  clamp  and  the  piece  D,  keeps 
them  at  a  fixed  distance  apart,  so  that  they  cannot  move  to  or  from 
one  another,  nor  consequently  the  two  circles  to  which  they  are 
respectively  made  fast.  But  when  this  tangent-screw  is  turned  by 
its  milled-head,  it  gives  the  clamp  and  with  it  the  upper  plate  a 
gmooth  and  slow  motion,  backward  or  forward,  whence  it  is  called 
the  "  Slow  motion  screw,"  as  well  as  "  Tangent-screw,"  from  the 
direction  in  which  it  acts.  It  is  always  placed  at  the  south  end 
of  the  compass-box. 

A  little  different  arrangement  is  employed  to  give  a  similar 
motion  to  the  lower  circle  (which  we  have  hitherto  regarded  as 
immovable)  on  the  body  of  the  instrument.  Its  axis  is  embraced 
by  a  brass  ring,  into  which  enters  another  tangent-screw,  which 
also  passes  through  a  piece  fastened  to  the  plate  P.  The  clam]> 
screw,  C,  causes  the  ring  to  pinch  and  hold  immovably  the  axis  of 
the  lower  circle,  while  a  turn  of  the  Tangent-screw,  T,  will  slow^ly 
move  the  clamp  ring  itself,  and  therefore  with  it  the  lower  circle. 
When  the  clamp  is  loosened,  the  lower  circle,  and  Avith  it  every 
thing  above  it,  has  a  perfectly  free  motion.  A  recent  improvement 
is  the  employment  for  this  purpose  of  two  tangent  screws,  pressing 
against  opposite  sides  of  a  piece  projecting  from  the  clamp-ring. 
One  is  tightened  as  the  other  is  loosened,  and  a  very  steady  mo- 
tion is  thus  obtained. 


(337)  Levels.  Since  the  object  of  the  instrument  is  to  measure 
horizontal  angles,  the  circular  plate  on  which  they  are  measured 
must  itself  be  made  horizontal.  \Yhether  it  is  so  or  not  is  known 
by  means  of  two  small  levels  placed  on  the  plate  at  right  angles  to 
each  other.  Each  consists  of  a  glass  tube,  slightly  cui'ved  upward 
in  its  middle  and  so  nearly  filled  with  alcohol,  that  only  a  small 
bubble  of  air  is  left  in  the  tube.  This  always  rises  to  the  highest 
part  of  the  tubes.  They  are  so  "adjusted"  (as  will  be  explained 
in  chapter  III)  that  when  this  bubble  of  air  is  in  the  middle  of 
the  tubes,  or  its  ends  equidistant  from  the  central  mark,  the  p!at« 


JSHAP.  I.] 


The  Instruments. 


225 


on  which  they  are  fastened  shall  be  level,  which  way  soever  it  may 
be  turned. 

The  levels  are  represented  in  the  figure  of  the  Transit,  on  page 
212,  as  being  under  the  plate.  They  are  sometimes  placed  above 
it.  In  that  case,  the  Verniers  are  moved  to  one  side,  between  the 
feet  of  the  standards,  and  one  of  the  levels  is  fixed  between  the 
standards  above  one  of  the  Verniers,  and  the  other  on  the  plate  at 
the  south  end  of  the  compass-box. 

(338)  Parallel  Plates*  To  raise  or  lower  either  side  of  the 
circle,  so  as  to  brmg  the  bubbles  into  the  centres  of  the  tubes, 
requires  more  gentle  and  steady  movements  than  the  unaided  hands 
can  give,  and  is  attained  by  the  Parallel  Plates  P,  P',  (so  called 
because  they  are  never  parallel  except  by  accident),  and  their 
four  screws  Q,  Q,  Q,  Q,  which  hold  the  plates  firmly  apart,  and,  by 
being  turned  in  or  out,  raise  or  lower  one  side  or  the  other  of  the 
upper  plate  P',  and  thereby  of  the  graduated  circle.  The  two 
plates  are  held  together  by  a  ball  and  socket  joint.  To  level  the 
instrument,  loosen  the  lower  clamp  and  turn  the  circle  till  each 
level  is  parallel  to  the  vertical  plane  passing  through  a  pair  of 
opposite  screws.  Then  take  hold  of  two  opposite  screws  and  turn 
them  simultaneously  and  equally,  but  in  contrary  directions,  screw- 

Fi2.  223. 


ing  one  in  and  the  other  out,  as  shown  by  the  aiTOws  in  the  figures. 
A  rule  easily  remembered  is  that  both  thumbs  must  turn  in,  or  both 
out.  The  movements  represented  in  the  first  of  these  figures  would 
raise  the  left-hand  side  of  the  circle  and  lower  the  right-hand  side, 
xhe  movements  of  the  second  figure  would  produce  the  reverse 
eflfect.  Care  is  needed  to  turn  the  opposite  screws  equally,  so  that 
they  shall  not  become  so  loose  that  the  instrument  wiU  rock,  or  so 
tight  as  to  be  cramped.  When  this  last  occurs,  one  of  the  other 
pair  should  be  loosened. 

15 


223        TRANSIT  AND  THEODOLITE  SURTEYIXG.       [part  it 

Sometimes  one  of  each  pair  of  the  screws  is  replaced  by  a  strong 
spring  against  which  the  remaining  screws  act. 

The  French  and  German  instrmnents  are  usually  supported  by 
only  three  screws.  In  such  cases,  one  level  is  brought  parallel  to 
one  pair  of  screws  and  levelled  by  them,  and  the  other  level  has 
its  bubble  brought  to  its  centre  by  the  third  screw.  If  there  is 
only  one  level  on  the  instrument,  it  is  first  brought  parallel  tc  one 
pair  of  screws  and  levelled,  and  is  then  turned  one  quarter  around 
so  as  to  be  perpendicular  to  them  and  over  the  third  screw,  and  the 
operation  is  repeated. 

(339)  Watch  Telescope.  A  second  Telescope  is  sometimes 
attached  to  the  lower  part  of  the  instrument.  When  a  number  of 
angles  are  to  be  observed  from  any  one  station,  direct  the  upper 
and  principal  Telescope  to  the  first  object,  and  then  direct  the 
lower  one  to  any  other  well-defined  point.  Then  make  all  the 
desired  observations  with  the  upper  Telescope,  and  when  they  are 
finished,  look  again  through  the  lower  one,  to  see  that  it  and  there- 
fore the  divided  circle  has  not  been  moved  by  the  movements  of 
the  Vernier  plate.  The  French  call  this  the  Witness  Telescope, 
(^Lunette  temoin). 

(340)  The  Compass.  Upon  the  upper  plate  is  fixed  a  compass. 
Its  use  has  been  fully  explained  in  Part  III.  It  is  fittle  used  in 
connection  with  the  Transit  or  Theodolite,  which  are  so  incompara- 
bly more  accurate,  except  as  a  "  check,"  or  rough  test  of  the 
accuracy  of  the  angles  taken,  which  should  about  equal  the  differ- 
ence of  the  magnetic  bearings.  Its  use  will  be  farther  noticed  in 
Chapter  IV,  on  "  Field  Work." 

(341)  The  Surveyor's  Transit.  lu  this  instrument  (so 
named  by  its  introducers,  Messrs.  Gurley,  and  shown  in 
Fig.  224),  the  Vernier-plate,  which  carries  the  standards 
and  telescope,  is  under  the  plate  which  carries  the  grad- 
uated circle,  and  the  compass  is  attached  to  the  latter. 
By  this  arrangement,  when  the  Vernier  is  set  at  any  angle, 
ihe  line  of  sight  of  the  telescope  will  make  that  angle  with 
the  IN",  and  S.  lines  of  the  compass.  Consequently,  this 
instrument  can  be  used  precisely  like  the  Vernier  compass 


CHAP.  I.] 


The  Instruments. 


227 


to    allow   for  magnet-  ^'s-  '^-^i- 

ic   variation,  and   thus 

to  run  out  a  line  with 

true  bearings,  as  in  Art. 

(312),  or  to  run  out  old 

lines,  allowing  for  the 

secular  variation,  as  in 

Art.  (321). 

The  instrument  may 
also  be  used  like  the 
commonEngineer's 
Transit.  The  compass, 
however,  will  then  not 
orive  the  bearins^s  of  the 
lines  surveyed,  but  they 
can  easily  be  deduced 
from  that  of  any  one 
line. 

(342)  Goniasmometre.  A  very  compact  in- 
strument to  Avhich  the  above  name  has  been 
given  in  France,  where  it  is  much  used,  is  shown 
in  the  figure.  The  upper  half  of  the  cyhnder  is 
movable  on  its  lower  half.  The  observations 
may  be  taken  through  the  shts,  as  in  the  Survey- 
or's Cross,  or  a  Telescope  may  be  added  to  it. 
Readings  may  be  taken  both  from  the  compass, 
and  from  the  divided  edge  of  the  lower  half  of 
the  cyhnder,  by  means  of  a  Vernier  on  the 
upper  half.* 


*  The  proper  care  of  instruments  mnst  not  be  overlooked. 
If  varnished,  they  should  be  wiped  gently  with  fine  and 
clean  linen.  If  polished  with  oil,  they  should  be  rubbed 
more  strongly.  The  parts  neither  varnished  nor  oiled,  should 
be  cleaned  with  Spanish  white  and  alcohol.  Varnished  wood,  when  spoltt-d. 
should  be  wiped  with  very  soft  linen,  moistened  with  a  little  olive  oil  or  alcohol, 
['npainted  wood  is  cleaned  with  sand-paper.  Apply  olive  oil  where  steel  ruhi 
against  brass ;  and  wax  softened  by  tallow  where  biass  rubs  against  brass. — 
Jlean  the  glasses  with  kid  or  buck  skin.     Wash  them,  if  dirtied,  with  alcohol. 


Fig.  224^ 


228 


f PART  !▼ 


CHAPTER  11. 


VEMIERS. 

(343)  Definition.  A  Vernier  is  a  contrivance  for  measuring 
smaller  portions  of  space  than  those  into  which  a  line  is  actually 
divided.  It  consists  of  a  second  line  or  scale,  movable  by  the  side 
of  the  first,  and  divided  into  equal  parts,  which  are  a  very  little 
shorter  or  longer  than  the  parts  into  which  the  first  line  is  divided. 
Ttis  small  difference  is  the  space  which  we  are  thus  enabled  tn 
measure.* 

The  Vernier  scale  is  usually  constructed  by  taking  a  length 
equal  to  any  number  of  parts  on  the  divided  line,  and  then  dividing 
this  length  into  a  number  of  equal  parts,  one  more  or  one  less  than 
the  number  into  which  the  same  length  on  the  original  line  is  di- 
vided. 


(344)  Illustration.  The  figure  represents  (to  twice  the  real 
size)  a  scale  of  inches  divided  into  tenths,  with  a  Vernier  scale 
beside  it,  by  which  hundredths  of  an  inch  can  be  measured.     The 


Fig.  225. 


/ 


2^®5NC^^'^C^<?^P— lO 


^ 


C^  t  CO 

Vernier  is  made  by  setting  off  on  it  9  tenths  of  an  inch,  and  divid- 
ing that  length  into  10  equal  parts.  Each  space  on  the  Vernier 
13  therefore  equal  to  a  tenth  of  nine-tenths  of  an  inch,  or  to  nine- 
hundredths  of  an  inch,  and  is  consequently  one-hundredth  of  an 
inch  shorter  than  one  of  the  divisions  of  the  original  scale.     The 

•  The  Vernier  is  so  named  from  its  inventor,  in  lfi31.  The  name  "  Nonius," 
often  improperly  given  to  it,  belongs  to  an  entirely  different  coniJivance  for  a 
similar  object. 


CHAP.  II.] 


Verniers. 


228 


first  space  of  the  Vernier  will  therefore  fall  short  of,  or  be  over- 
lapped bj,  the  first  space  on  the  scale  by  this  one-hundredth  of  an 
inch ;  the  second  space  of  the  Vernier  will  fall  short  by  tAvo-hun 
dreiths  of  an  inch;  and  so  on.  If  then  the  Vernier  be  moved  up 
by  the  side  of  the  original  scale,  so  that  the  line  marked  1  coin- 
cides, or  forms  one  straight  line,  with  the  line  of  the  scale  which 
was  just  above  it,  we  know  that  the  Vernier  has  been  moved  one- 
hundredth  of  an  inch.  If  the  hne  marked  2  comes  to  coincide 
with  a  line  of  the  scale,  the  Vernier  has  moved  up  two-hundredths 
of  an  inch ;  and  so  for  other  numbersi     If  the  position  of  the 

Fig.  226. 


1 

\             Y 

... 

1        1 

< 

\ 

C^ 

t 

CO 

Vernier  be  as  in  this  figure,  the  line  marked  7  on  the  Vernier 
corresponding  Avith  some  line  on  the  scale,  the  zero  line  of  the 
Vernier  is  7  hundredths  of  an  inch  above  the  division  of  the  scale 
next  below  this  zero  line.  If  this  division  be,  as  in  the  figure, 
8  inches  and  6  tenths,  the  reading  wiU  be  8.67  inches.* 

A  Vernier  like  this  is  used  on  some  levelling  rods,  being  engraved 
on  the  sides  of  the  opening  in  the  part  of  the  target  above  its 
middle  line.  The  rod  being  divided  into  hundredths  of  a  foot,  tins 
Vernier  reads  to  thousandths  of  a  foot.  It  is  also  used  on  some 
French  Mountain  Barometers,  which  are  divided  to  hundredths  of 
a  metre,  and  thus  read  to  thousandths  of  that  unit. 

(345)  General  rules.  To  find  what  any  Vernier  reads  to, 
i.  e.  tc  determine  how  small  a  distance  it  can  measure,  observe 
how  many  parts  on  the  original  line  are  equal  to  the  same  number 
mcreased  or  diminished  by  one  on  the  Vernier,  and  divide  the 


*  The  student  will  do  well  to  draw  such  a  scale  and  Vernier  on  two  slips  of 
thick  paper,  and  move  one  beside  the  other  till  he  can  read  them  iii  any  possible 
position  ;  and  so  with  the  following  Verniers. 


230        TRANSIT  AXD  THEODOLITE  SURVEYING.       [part  it 

length  of  a  part  on  the  original  line  by  this  last  number.  It  will 
give  the  required  distance.* 

To  read  any  Vernier^  firstly,  look  at  the  zero  line  of  the  Ver 
nier,  (which  is  sometimes  marked  bj  an  arrow-head),  and  if  it 
coincides  with  any  division  of  the  scale,  that  wiU  be  the  correct 
reading,  and  the  Vernier  divisions  are  not  needed.  But  if,  as 
usually  happens,  the  zero  line  of  the  Vernier  comes  between  any 
two  divisions  of  the  scale,  note  the  nearest  next  less  division  on  the 
scale,  and  then  look  along  the  Vernier  till  you  come  to  some  hne 
on  it  which  exactly  coincides,  or  forms  a  straight  line,  with  some 
line  (no  matter  which)  on  the  fixed  scale.  The  number  of  this 
line  on  the  Vei'nier  (the  7th  in  the  last  figure)  tells  that  so  many 
of  the  sub-divisions  which  the  Vernier  indicates,  are  to  be  added  to 
the  reading  of  the  entire  divisions  on  the  scale. 

When  several  lines  on  the  Vernier  appear  to  coincide  equally 
with  lines  of  the  scale,  take  the  middle  line. 

When  no  line  coincides,  but  one  Hne  on  the  Vernier  is  on  one 
side  of  a  line  on  the  scale,  and  the  next  line  on  the  Vernier  is  as 
far  on  the  other  side  of  it,  the  true  reading  is  midway  between  those 
indicated  by  these  two  lines. 

(346)  Retrograde  Verniers.  The  spaces  of  the  Vernier  in 
modern  instruments,  are  usually  each  shorter  than  those  on  the  scale, 
a  certain  number  of  parts  on  the  scale  being  divided  into  a  larger 
number  of  parts  on  the  Vernier. f  In  the  contrary  case,|  there  is 
the  inconvenience  of  being  obliged  to  number  the  lines  of  the  Ver- 
nier and  to  count  their  coincidences  with  the  lines  of  the  scale,  in 
a  retrograde  or  contrary  direction  to  that  in  which  the  numbers  on 
the  scale  run    We  will  call  such  arrangements  retrograde  Verniers. 

•  In  Algebraic  language,  let  s  equal  the  length  of  one  part  on  the  original  line, 
and  V  the  unknown  length  of  one  part  on  the  Vernier.     Let  m  of  the  former  = 

m  -)-  1  of  the  latter      Then  ms  =  (m  -ir-  I)  v.        v  = s.        «  —  t»  =  «  — 


«  +  1         m  -\-  i 

*  i  e.  Algebraically,  v  = t  i.  e.  When  v  = «, 

m  -j-  1  m  —  1 


''     I 


C  (,■!- 


'ry^^^P^'^'^c^  ^O  ^ 


yuS^ 


CHAP.  II.J 


Verniers. 


231 


(347)  Illustration.  In  this  figure,  the  scale,  as  before,  repre- 
Bents  (to  twice  the  real  size)  inches  divided  into  tenths,  but  the 
Vernier  is  made  bj  dividing  11  parts  of  the  scale  into  10  equal 

Fig.  227. 


\ 


o^<rvf<W'rH^wt>«coct>2 


Y_ 


parts,  each  of  which  is  therefore  one-tenth  of  eleven-tenths  of  ai 
inch,  i.  e.  eleven-hundredths  of  an  inch,  or  a  tenth  and  a  hun- 
dredth. Each  space  of  the  Vernier  therefore  overlaps  a  space  on 
the  scale  by  one-hundredth  of  an  inch.  The  manner  of  reading 
this  Vernier  is  the  same  as  in  the  last  one,  except  that  the  numbers 
run  in  a  reverse  direction.     The  reading  of  the  figure  is  30.16. 

This  Vernier  is  the  one  generaUj  apphed  to  the  common  Baro- 
meter, the  zero  point  of  the  Vernier  being  brought  to  the  level  of 
the  top  of  the  mercury,  whose  height  it  then  measures.  It  is  also 
employed  for  levelling  rods  which  read  downwards  from  the  middle 
of  the  target. 


(348)  The  figure  below  represents  (to  double  size)  the  usual 
scale  of  the  English  Mountain  Barometer.*  The  scale  is  first 
divided  into  inches.     These   are  subdivided  mto  tenths  by  the 

Fig.  228. 


kJp         ^ 


(TO 


% 


I   I  I  I  I  I  I  I  I  IT 


I     I      r 


St 


CTi 


'  This  figure,  and  others  in  this  chapter,  are  from  Bree's  "  Present  Practice." 


232 


TRANSIT  AND  THEODOLITE  SURVEYING.    I  part  iv 


longer  lines,  and  the  shorter  lines  again  divide  these  into  half 
tenths,  or  to  5  hundredths.  24  of  these  smaller  parts  are  set  off 
on  the  Vernier,  and  divided  into  25  equal  parts,  each  of  which  ia 


therefore  = 


24  X  .05 
25 


=  .048  inch,  and  is  shorter  than  a  division 


of  the  scale  by  .050  —  .048  =  .002,  or  two  thousandths  of  an  inch, 
a  twenty-fifth  part  of  a  division  on  the  scale,  to  which  minuteness 
the  Vernier  can  therefore  read.  The  i-ttading  in  the  figure  is 
£0.686,  (30.65  by  the  scale  and  .036  by  the  Vernier),  the  dotted 
line  marked  D  showing  where  the  coincidence  takes  place. 

(349)  Circle  divided  into  degrees.  The  following  illustra- 
tions apply  to  the  measurements  of  angles,  the  circle  being  vari- 
ously divided.  In  this  article,  the  circle  is  supposed  to  be  divided 
into  degrees. 

If  6  spaces  on  the  Vernier  are  found  to  be  equal  to  5  on  the 
circle,  the  Vernier  can  read  to  one-sixth  of  a  space  on  the  circle, 
i.  e.  to  10'. 

If  10  spaces  on  the  Vernier  are  equal  to  9  on  the  circle,  the 
Vernier  can  read  to  one-tenth  of  a  space  on  the  circle,  i.  e.  to  6'. 

If  12  spaces  on  the  Vernier  are  equal  to  11  on  the  circle,  the 
Vernier  can  read  to  one-twelfth  of  a  space  on  the  circle,  i.  e.  to  5'. 


10 


Fig.  229. 


i 


60    45     30 


The  above  figure  shows  such  an  arrangement.  The  index,  or 
zero,  of  the  Vernier  is  at  a  point  beyond  358°,  a  certain  distance, 
which  the  coincidence  of  the  third  Ime  of  the  Vernier  (as  indicated 


£HAP.  II.] 


Verniers, 


233 


by  the  dotted  and  crossed  line)  shows  to  be  15'.  The  whole  read- 
mg  is  therefore  358°  15'. 

If  20  spaces  on  the  Vernier  are  equal  to  19  on  the  circle,  the 
Vernier  can  read  to  one-twentieth  of  a  division  on  the  circle, 
i.  e.  to  3'.  EngUsh  compasses,  or  "  Circumferentors,"  are  some- 
times thus  arranged. 

If  60  spaces  on  the  Vernier  are  equal  to  59  on  the  circle,  the 
Vernier  can  read  to  one-sixtieth  of  a  division  on  the  circle,  i.  e.  to  1'. 


(350)   Circle  divided  to  3j'.     Such  a  graduation  is   a  very 
common  one.     The  Vernier  may  be  variously  constructed. 

Suppose  30  spaces  on  the  Vernier  to  be  equal  to  29  on  the 

29  X  30' 


circle.      Each  space  on  the  Vernier  will  be 


30 


—  =  29', 


and  will  therefore  be  less  than  a  space  of  the  circle  by  1',  to  which 
the  Vernier  will  then  read. 


F;l-.  230. 


The  above  figure  shows  tliis  arrangement.  The  reading  is  0°, 
or  360°. 

In  the  followino;  fiorure,  the  dotted  and  crossed  line  shows  what 
divisions  coincide,  and  the  reading  is  20°  10';  the  Vernier  being 
the  same  as  in  the  preceding  figure,  and  its  zero  being  at  a  point 
of  the  circle  10'  beyond  20 \ 


234        TRANSIT  AI^D  THEODOLITE  SURVEYIIVG.      [parc  iv. 

Fis;.  231. 


In  the  following  figure,  the  reading  is  20°  40',  the  index  being 
at  a  point  beyond  20°  30',  and  the  additional  space  being  shown 
by  the  Vernier  to  be  10'. 


Fig.  232. 


CUAP.  II. 1 


Verniers. 


235 


Sometimes  30  spaces  on  the  Vernier  are  equal  to  31  on  the  cu-cle, 

31  X  30' 
Each  space  on  the  Vernier  will  therefore  be  = ~ —  =  31',  and 

will  be  longer  than  a  space  on  the  circle  by  1',  to  which  it  will 
therefore  read,  as  in  the  last  case,  but  the  Vernier  will  be  "  retro- 
grade."    This  is  the  Vernier  of  the  compass,  Fig.  148.     The  pecu- 
liar manner  in  which  it  is  there  applied  is  sho^-n  in  Fig.  239. 
If  15  spaces  on  the  Vernier  are  equal  to  16  on  the  circle,  each 

16  X  30' 
space  on  the  Vernier  will  be  =  — ij-;^ —  =  32',  and  the  Vernier 


15 


will  therefore  read  to  2'. 


^ 


(351)  Circle  divided  to  20'.  If  20  spaces  on  the  Vernier 
are  equal  to  19  on  the  circle,  each  space  of  the  latter  will  be  = 
19  X  20'  ^  ^^,^  ^^^  ^^^  Vernier  will  read  to  20'—  19'  =  1'. 

If  40  spaces  on  the  Vernier  are  equal  to  41  on  the  circle,  each 

41  X  20' 

=  20V;andtheVer- 


space  on  the  Vernier  will  be  = 
nier  will  therefore  read  to  20^' 


40  ^  ' 

20'  =  30".  It  wiU  be  retro- 
grade. In  the  following  figure  the  reading  is  360",  or  0°  ;  and  it 
will  be  seen  that  the  40  spaces  on  the  Vernier  (numbered  to  whole 
minutes)  are  equal  to  13°  40'  on  the  limb,  i.  e.  to  41  spaces,  each 
of  20'. 

Fk.233. 


If  60  spaces  on  the  Vernier  are  equal  to  59  on  the  circle,  each 

59  X  '^0' 
of  the  former  will  be  =  — ^-^-  =  19'  40",  and  the  Vernier 

bU 


236        TRAi\SIT  AlVD  THEODOLITE  SURVEYING.      [part  iv 

will  therefore  read  to  20'  — 19'  40"  =  20".  The  foUowmg  figure 
shows  such  an  arrangement.  The  reading  in  that  position  -would 
be  40'  46'  20". 


Fig.  Q34. 


45 


40 


W  9    8    ?  !6    5   4    3    2    I 


--V- 


r^^/\f 


/i''"- 


I 


(352)  Circle  dirided  to  15'.     If  60  spaces  on  the  Vernier  are 

equal  to  59  on  the  circle,  each  space  on  the  Vernier  will  be  = 

59  X  15' 

=  14'  45",  and  the  Vernier  will  read  to  15".     In  the 


60 


following  figure  the  reading  is  10°  20'  45",  the  index  pointing  to 
10°  15',  and  something  more,  which  the  Vernier  shows  to  be  5'  45' 

Fig.  235. 


15 


10 


I  I 


'III 


W  9  8   7  6:  5  4    3  2   1 


/'|\ 


f 


CHAH.  II.] 


Verniers. 


237 


(353)  Circle  divided  to  10'.  If  60  spaces  on  the  Vernier  be 
equal  to  59  on  the  limb,  the  Vernier  will  read  to  10".  In  the 
following  figure,  the  reading  is  7°  25'  40",  the  reading  on  the 
circle  being  7°  20',  and  the  Vernier  showing  the  remaining  space 
to  be  5'  40". 

Fi^.  236. 


10  9   8  7  6 


5  4  3    2   1 


y^S 


\ 


C354)  Readmi?  backwards.  AVhen  an  index  carrying  a  Ver-' 
nier  is  moved  backwards,  or  in  a  contrary  direction  to  that  in 
which  the  numbers  on  the  circle  run,  if  we  wish  to  read  the  spa^e 
which  it  has  passed  over  in  this  du-ection  from  the  zero  point,  the 
Vernier  must  be  read  backwards,  (i.  e.  the  highest  number  be 
called  0),  or  its  actual  reading  must  be  subtracted  from  the  value 
of  the  smallest  space  on  the  cii'cle.  The  reason  is  plain  ;  for, 
since  the  Vernier  shows  how  far  the  index,  moving  in  one  direc- 
tion, has  gone  past  one  di\ision  line,  the  distance  which  it  is  from 
the  next  di\dsion  line  (wliich  it  may  be  supposed  to  have  passed, 
moving  in  a  contrary  direction),  TviU  be  the  difference  between  the 
reading  and  the  value  of  one  space. 

Thus,  in  Fig.  229,  page  232,  the  reading  is  358°  15'.     But, 
counting  backwards  from  the  360°,  or  zero  point,  it  is  1°  45'. 

Caution  on  this  point  is  particularly  necessary  in  using  siT»aIl 
angles  of  deflection  for  railroad  curves. 


238  TRAXSIT  AXD  THEODOLITE  SIIIIFE¥IXG.     [part  it. 


(355)  Arc  of  excess.  On  the  sextant  and  similar  instru- 
ments, the  divisions  of  the  limb  are  carried  onward  a  short  distance 
beyond  the  zero  point.  This  portion  of  the  limb  is  called  the  "Arc 
of  excess."  When  the  index  of  the  Vernier  points  to  this  arc,  the 
reading  must  be  made  as  explained  in  the  last  article.  Thus,  in 
the  figure,  the  reaiing  on  the  arc  from  the  zero  of  the  limb  to  the 

Fia.  23" 


10  9   8  7 


6    5  4  3  Z  1 


zero  of  the  Vernier  is  4°  20',  and  something  more,  and  the  reading 
of  the  Vernier  from  10  towards  to  the  right,  where  the  lines  coin- 
cide, is  3'  20",  (or  it  is  10'  —  6'  40"  =  3'  20"),  and  the  entire 
reading  is  therefore  4°  23'  20". 

(356)  Double  Verniers.  To  avoid  the  inconveniences  of  read- 
ing backwards,  double  Verniers  are  sometimes  used.  The  figure 
below  shows  one  appHed  to  a  Transit.     Each  of  the  Verniers  is 

Fiff.  238. 


30jJ0j_WjJ/  ♦  10  ♦  20 


OHAP.  II.] 


Verniers. 


239 


like  the  one  described  in  Art.  (350),  Figs.  230,  231,  and  282. 
When  the  degrees  are  counted  to  the  left,  or  as  the  numbers  run, 
as  is  usual,  the  left-hand  Vernier  is  to  be  read,  as  in  Art.  (3,50)  ; 
but  when  the  degrees  are  counted  to  the  right,  from  the  860°  line, 
the  risht-hand  Vernier  is  to  be  used. 


(357)  Compass-Vernier,     Another  form  of  double  Vernier, 
often  applied  to  the  compass,  is  shown  in  the  following  figure.    The 


Fisr.  239. 


2o 


30 


25 


f. 


10    5    4    5    i!o 


0 


I 


limb  is  divided  to  half  degrees,  and  the  Vernier  reads  to  mmutes, 
30  parts  on  it  being  equal  to  31  on  the  limb.  But  the  Vernier  is 
only  half  a-s  long  as  in  the  previous  case,  going  only  to  15',  the 
upper  figures  on  one  half  being  a  sort  of  continuation  of  the  lower 
figures  on  the  other  half.  Thus  in  moving  the  index  to  the  right, 
read  the  lozver  figures  on  the  left  hand  Vernier  (it  being  retro- 
grade) at  any  coincidence,  when  the  space  passed  over  is  less  than 
15' ;  but  if  it  be  more,  read  the  upper  figures  on  the  right  hand 
Vernier :  and  vice  versa. 


240  [part  it 


CHAPTER  III. 


ADJUSTMENTS. 

(358)  The  purposes  for  which  the  Transit  and  Theodolite  (as  well 
as  most  surveying  and  astronomical  instruments)  are  to  be  used, 
require  and  presuppose  certain  parts  and  lines  of  the  instrument 
to  be  placed  in  certain  directions  with  respect  to  others  ;  these  re- 
spective directions  being  usually  parallel  or  perpendicular.  Such 
arrangements  of  their  parts  are  called  their  Adjustments.  The 
same  word  is  also  applied  to  placmg  these  lines  in  these  directions. 
In  the  following  explanations  the  operations  which  determine 
whether  these  adjustments  are  correct,  will  be  called  their  Verifi- 
cations ;  and  the  making  them  right,  if  they  are  not  so,  their  Hec- 
tifications* 

(359)  In  observations  of  horizontal  angles  with  the  Transit  or 
the  Theodohte,f  it  is  required, 

1°  That  the  circular  plates  shall  be  horizontal  in  whatever  way 
they  may  be  turned  around. 

2°  That  the  Telescope,  when  pointed  forward,  shall  look  in  pre- 
cisely the  reverse  of  its  direction  when  pointed  backward,  i.  e.  that 
its  two  lines  of  sight  (or  lines  of  collimation)  forward  and  back- 
ward shall  lie  in  the  same  plane. 

3°  That  the  Telescope  in  turning  upward  or  downward,  shall 
move  in  a  truly  vertical  plane,  in  order  that  the  angle  measured 
between  a  low  object  and  a  high  one,  may  be  precisely  the  same 
as  would  be  the  angle  measured  between  the  low  object  and  a  point 
exactly  under  the  high  object,  and  in  the  same  horizontal  plane  as 
the  low  one. 

*  It  has  been  well  said,  tliai  "  in  the  present  state  of  science  it  may  be  laid 
down  as  a  maxim,  that  eveiy  instniment  should  be  so  conti'ived,  that  the  observer 
may  easily  examine  and  rectify  the  principal  parts  ;  for,  however  careful  the 
instrument-maker  may  be,  however  perfect  the  execution  thereof,  it  is  not  possi- 
ble that  any  instrument  should  long  remain  accurately  fixed  in  the  position  in 
which  it  came  out  of  the  maker's  hands." — Adams'  "  Geometrical  and  GrarJiical 
Essays,"  1791. 

t  The  Theodolite  adjustments  which  relate  only  to  levelling  or  to  measuring 
vertical  angles,  will  not  be  here  discussed. 


CHAP,  iii-l  Adjustments.  241 

We  shall  see  that  all  these  adjustments  are  finally  resolvable 
into  these  ;  1st,  Making  the  vertical  axis  of  the  instrument  perpen- 
dicular to  the  plane  of  the  levels ;  2d.  Makmg  the  line  of  collima- 
tion  perpendicular  to  its  axis ;  and  3d.  Making  this  axis  parallel 
to  the  plane  of  the  levels.  They  are  all  best  tested  by  the  invalu* 
able  princ^le  of  "  Reversion." 

We  have  now,  firstly,  to  examine  whether  these  things  are  so, 
that  is,  to  "  verify"  the  adjustments ;  and,  secondly,  if  we  find  that 
they  are  not  so,  to  malce  them  so,  i.  e.  to  "  rectify,"  or  "  adjust"  them 
correctly.  The  above  three  requirements  produce  as  many  corre- 
sponding adjustments. 

(360)  First  adjustment.  To  cause  the  circle  to  he  horizontal 
in  every  position* 

Verification. — Turn  the  Vernier  plate  which  carries  the  levels, 
till  one  of-  them  is  parallel  to  one  pair  of  the  parallel  plate 
screws.  The  other  will  then  be  parallel  to  the  other  pair.  Bring 
eaish  bubble  to  the  middle  of  its  tube,  by  that  pair  of  screws  to 
which  it  is  parallel.  Then  turn  the  vernier  plate  half  way  around, 
i.  e.  till  the  index  has  passed  over  180°.  If  the  bubbles  remain 
in  the  centres  of  the  tubes,  they  are  in  adjustment.  If  either  of 
them  runs  to  one  end  of  the  tube,  it  requires  rectification. 

Rectification. — The  fault  which  is  to  be  rectified  is  that  the 
plane  of  the  level  (i.  e.  the  plane  tangent  to  the  highest  point  of 
the  level  tube)  is  not  perpendicular  to  the  vertical  axis,  AA  in 
figure  214,  on  which  the  plate  turns.     For,  let  AB  represent  this 

Fig.  240.  Fig.  241. 


plane,  seen  edgeways,  and  Ct)  the  centre  line  of  the  vertical  axis, 

This  applies  equally  to  the  Transit  and  the  Theodobte. 

16 


342        TRANSIT  AND  THEODOLITE  SrRVEYL\G.      [part  it 

which  is  here  drawn  as  making  an  acute  angle  with  this  plane 
9n  the  right  hand  side.  The  first  figure  represents  the  bubhle 
brought  to  the  centre  of  the  tube.  The  second  figure  represents 
the  plate  turned  half  around.  The  centre  Une  of  the  axis  is  sup- 
posed  to  remain  unmoved.  The  acute  angle  will  now  be  on  the 
left  hand  side,  and  the  plate  will  no  longer  be  horizontal.  Conse- 
quently the  bubble  will  run  to  the  higher  end  of  the  tube.  The 
rectification  necessary  is  evidently  to  raise  one  end  of  the  tube  and 
lower  the  other..  The  real  error  has  been  doubled  to  the  eye  by 
the  reversion.  Half  of  the  motion  of  the  bubble  was  caused  by  the 
tangent  plane  not  being  perpendicular  to  the  axis,  and  half  by  this 
axis  not  being  vertical.  Therefore  raise  or  lower  one  end  of  the 
level  by  the  screws  which  fasten  it  to  the  plate,  till  the  bubble 
comes  about  half  way  back  to  the  centre,  and  then  bring  it  quite 
back  by  turning  its  pair  of  parallel  plate  screws.  Then  again 
reverse  the  vernier  plate  180°.  The  bubble  should  now  remain  in 
the  centre.  If  not,  the  operation  should  be  repeated.  The  same 
must  be  done  with  the  other  level  if  required.  Then  the  bubbles 
will  remain  in  the  centre  during  a  complete  revolution.  This 
proves  that  the  axis  of  the  vernier  plate  is  then  vertical ;  and  as 
it  has  been  fixed  by  the  maker  perpendicular  to  the  plate,  the 
latter  must  then  be  horizontal. 

It  is  also  necessary  to  examine  whether  the  bubbles  remain  in 
the  centre,  when  the  divided  circle  is  turned  round  on  its  axis. 
K  not,  the  axes  of  the  two  plates  are  not  parallel  to  each  other. 
The  defect  can  be  remedied  only  by  the  maker ;  for  if  the  bubbles 
be  altered  so  as  to  be  right  for  this  reversal,  they  will  be  wrong 
for  the  vernier  plate  reversal 

(361)  Second  adjustment.  To  cause  the  line  of  collimation  to 
revolve  in  a  plane* 

Verifieation.  Set  up  the  Transit  in  the  middle  of  a  level  piece 
of  ground,  as  at  A  in  the  figure.  Level  it  carefully.  Set  a  stake, 
with  a  nail  driven  into  its  head,  or  a  chain  pin,  as  far  from  the 
instrument  as  it  is  distinctly  visible,  as  at  B.     Direct  the  telescope 

Ihis  adjustment  is  not  the  same  in  the  Transit  and  in  tlie  Theodolite.  Thai 
for  the  Transit  will  be  first  given,  and  that  for  the  Theodolite  iii  tlie  next  article 


CHAP.  III.] 


Adjustments. 


248 


Fig.  242 


to  it,  and  fix  the  intersection  of  the  cross-hairs  very  precisely  upoD 
it.  Clamp  the  instrument.  Measure  from  A  to  B.  Then  turn 
over  the  telescope,  and  set  another  stake  at  an  equal  distance  from 
the  Transit,  and  also  precisely  in  the  line  of  sight.  If  the  Ime  of 
collimation  has  not  continued  in  the  same  plane  during  its  half-revo- 
lution, this  stake  Avill  not  be  at  E,  but  to  one  side,  as  at  C.  To 
discover  the  truth,  loosen  the  clamp  and  turn  the  vernier  plate  half 
around  without  touching  the  telescope.  Sight  to  B,  as  at  first,  and 
again  clamp  it.  Then  turn  over  the  telescope,  and  the  line  of  sight 
will  strike,  as  at  D  in  the  figure,  as  far  to  the  right  of  the  point,  as 
it  did  before  to  its  left. 

Rectification.  The  fault  which  is  to  be  rectified,  is  that  the  line 
of  collimation  of  the  telescope  is  not  perpendicular  to  the  horizontal 
axis  on  which  the  telescope  revolves.     This  will  be  seen  by  the 

figures,  which  represent  the  position  of  the  lines  in  each  of  the  four 

A 


f  ig.  243.  B  — 


^-^ 


Fig.  244. 


Fig.  245.  B  — 


A. 


r 


Fig.  246. 


observations  which  have  been  made.  In  each  of  the  figures  the 
long  thick  line  represents  the  telescope,  and  the  short  one  the  axi^ 
on  which  it  turns.     In  Fig.  243  the  Une  of  sight  is  directed  to  B. 


241  TUAIVSIT  Ai\D  THEODOLITE  SURVEYOG.    [part  iv. 

In  Fig.  244  the  telescope  has  been  turned  over,  and  with  it  the 
axis,  so  that  the  obtuse  angle,  marked  0  in  the  first  figure,  has 
taken  the  place,  0',  of  the  acute  angle,  and  the  telescope  points  to 
C  instead  of  to  E.  In  Fig.  245  the  vernier  plate  has  been  turned 
half  around  so  as  to  point  to  B  again,  and  the  same  obtuse  angle 
has  got  around  to  0".  In  Fig.  246  the  telescope  has  been  turned 
over,  the  obtuse  angle  is  at  0'",  and  the  telescope  now  points  to  D. 

To  make  the  line  of  coUimation  perpendicular  to  the  axis,  the 
f -rmer  must  have  its  direction  changed.  This  is  effected  by  mov- 
ing the  vertical  hair  the  proper  distance  to  one  side.  As  was 
explained  in  Art.  (330),  and  represented  in  Fig.  217,  the  cross- 
hairs are  on  a  ring  held  by  four  screws.  By  loosening  the  left- 
hand  screw  and  tightening  the  right-hand  one,  the  ring,  and  with 
it  the  cross-hairs,  will  be  drawn  to  the  right ;  and  vice  veisa.  Two 
lioles  at  right  angles  to  each  other  pass  through  the  outer  heads  of 
the  screws.  Into  these  holes  a  stout  steel  wire  is  inserted,  and 
the  screws  can  thus  be  turned  around.  Screws  so  made  are  called 
'^  capstan-headed."  One  of  the  other  pair  of  screws  may  need  to 
be  loosened  to  avoid  strainuig  the  threads.  In  some  French  instru- 
ments, one  of  each  pair  of  screws  is  replaced  by  a  spring. 

To  find  how  much  to  move  this  vertical  hair,  measure  from  C  to 
D,  Fig.  242,  page  243.  Set  a  stake  at  the  middle  point  E,  and 
set  another  at  the  point  F,  midway  between  D  and  E.  Move  the 
vertical  hair  till  the  line  of  sight  strikes  F.  Then  the  instrument 
is  adjusted ;  and  if  the  line  of  sight  be  now  directed  to  E,  it  will 
strike  B,  when  the  telescope  is  turned  over ;  since  the  hair  is 
moved  half  of  the  doubled  error,  DE.  The  operation  will  gene- 
rally require  to  be  repeated,  not  being  quite  perfected  at  first. 

It  should  be  remembered,  that  if  the  Telescope  used  does  not 
invert  objects,  its  eye-piece  will  do  so.  Consequently,  with  such  a 
telescope,  if  it  seems  that  the  vertical  hair  should  be  moved  to  the 
left,  it  must  be  moved  to  the  right,  and  vice  versa.  An  inverting 
lelescope  does  not  invert  the  cross-hairs. 

If  the  young  surveyor  has  any  doubts  as  to  the  perfection  of  his 
rectification,  he  may  set  another  stake  exactly  under  the  instrument 
by  means  of  a  plumb-line  suspended  from  its  centre  ;  and  then,  in 
hke  manner,  set  his  Transit  over  B  or  E.     He  will  find  that  the 


CHAP.  Ill]  Adjustments.  2to 

other  two  stakes,  A  and  the  extreme  one,  are  in  the  same  straight 
line  with  his  instrument. 

In  some  instruments,  the  horizontal  axis  of  the  telescope  can  be 
taken  out  of  its  supports,  and  turned  over,  end  for  end.  In  such 
a  case,  the  line  of  sight  may  be  directed  to  any  well  defined  point, 
and  the  axis  then  taken  out  and  turned  over.  If  the  Une  of  sight 
again  strikes  the  same  point,  this  line  is  perpendicular  to  the  axis. 
If  not,  the  apparent  error  is  double  the  real  error,  as  appears  from 
the  figures,  the  obtuse  angle  0  coming  to  0',  and  the  desired  per- 

Fig.  247.  B ^2 %^ 


Fig.  24S    B "_-_":::r--;i5Si'^^0^ 

c —  ,,----' 

pendicular  Ime  falling  at  C  midway  between  B  and  B'.  The  rec- 
tification may  be  made  as  before ;  or,  in  some  large  instruments,  in 
which  the  telescope  is  supported  on  \8,  by  moving  one  of  the  Y^ 
laterally. 

(362)  The  Theodolite  must  be  treated  difierently,  since  its  tele- 
scope does  not  reverse.  One  substitute  for  this  reversal,  when  it 
is  desired  to  range  out  a  line  forward  and  backward  from  one  sta- 
tion, is,  after  sighting  in  one  direction,  to  take  the  telescope  out  of 
the  Ys  and  turn  it  end  for  end,  to  sight  in  the  reverse  direction. 
This  it  can  be  made  to  do  by  adjusting  its  line  of  collimation  as 
explained  in  the  last  article.  Another  substitute  is,  after  sighting 
in  one  direction,  and  noting  the  reading,  to  turn  the  vernier  plate 
around  exactly  180^.  But  this  supposes  not  only  that  the  gradua- 
tion is  perfectly  accurate,  but  also  that  the  line  of  collimation  is 
exactly  over  the  centre  of  the  circle.  To  test  this,  after  sighting 
to  a  point,  and  noting  the  reading,  take  the  telescope  out  of  the 
Ys  and  turn  it  end  for  end,  and  then  turn  the  vernier  plate 
around  exactly  180°.  If  the  line  of  sight  again  strikes  the  same 
point,  the  latter  condition  exists.     If  not,  the  maker  must  remedy 


246        TRAIVSIT  AAD  THEODOLITE  SFRVEYLXC.       [part  ir 

the  defect.     This  error  of  eccentricity  is  similar  to  that  explauaed 
with  respect  to  the  compass,  in  the  latter  part  of  Art.  (226). 


(303)   Third  adjustment.     To  cause  the  line  of  collimation 
revolve  in  a  vertical  plane  * 

Verification.  Suspend  a  long  pkTmb-line  from  some  high  point 
Set  the  instrument  near  this  line,  and  level  it  carefully.  Direct 
the  telescope  to  the  plumb-line,  and  see  if  the  intersection  of  the 
cross-hairs  follows  and  remains  upon  this  line,  when  the  telescope 
is  turned  up  and  down.     If  it  does,  it  moves  in  a  vertical  plane. 

The  angle  of  a  new  and  well-built  house  will  ftrm  an  imptrfect 
substitute  for  the  plumb-line. 

Otherwise;  the  instrument  being  set  up  and  levelled  as  above, 
place  a  basin  of  some  reflecting  liquid  (quicksilver  being  the  best, 
though  molasses,  or  oil,  or  even  water,  will  an'fewer,  though  less  per- 
fectly,) so  that  the  top  of  a  steeple,  or  other  point  of  a  high  object, 
can  be  seen  in  it  through  the  telescope  by  reflection.  Make  the 
intersection  of  the  cross-hairs  cover  it.  Then  turn  up  the  tele- 
scope, and  if  the  intersection  of  the  cross-hairs  bisects  also  the 
object  seen  directly,  the  hne  of  sight  has  moved  in  a  vertical  plane. 
If  a  star  be  taken  as  the  object,  'the  star  and  its  reflection  will  bo 
equivalent  (if  it  be  nearly  over  head)  to  a  plumb-line  at  least  fifty 
million  million  miles  long. 

Otherwise ;  set  the  instrument  as  close  as  possible  to  the  base 
of  a  steeple,  or  other  high  object ;  level  it,  and  direct     Fig-  249 
it  to  the  top  of  the  steeple,  or  to  some  other  elevated  ^ 

and  well  defined  point.  Clamp  the  plates.  Turn  down 
the  telescope,  and  set  up  a  pin  in  the  ground  pre- 
cisely "  in  line."  Then  loosen  the  clamp,  turn  over 
the  telescope,  and  turn  it  half-way  around,  or  so 
far  as  to  again  sight  to  the  high  point.  Clamp 
the  plates,  and  again  turn  down  the  telescope.  If 
the  line  of  sight  again  strikes  the  pin,  the  telescope 
has  moved  in  a  vertical  plane.  If  not,  the  apparent  5'  ^  ^' 
error  is  double  the  real  error.     For,  let  S  be  the  top  of  the  steeple, 

*  This  applies  to  both  the  Transit  and  the  Theodolite,  with  the  exception  of  the 
method  of  verification  by  the  steepl^  and  pin,  which  diiplies  only  to  the  Transit 


CHAP.  III.] 


Adjustments. 


247 


(Fig.  249)  and  P'  the  pin ;  then  the  plane  in  which 
the  telescope  moves,  seen  edgewise,  is  SP' ;  and, 
after  being  turned  around,  the  hne  of  sight 
moves  in  the  plane  SP",  as  far  to  one  side  of 
the  vertical  plane  SP,  as  SP'  was  on  the  other 
side  of  it. 

Rectification.  Since  the  second  adjustment 
causes  the  line  of  sight  to  move  in  a  plane  per- 
pendicular to  the  axis  on  which  it  turns,  it  will 
move  in  a  vertical  plane  if  that  axis  be  hori- 
zontal. It  may  be  made  so  by  filing  off  the 
feet  of  the  standards  which  support  the  higher 
end  of  the  axis.  This  will  be  best  done  by  the 
maker.  In  some  instruments  one  end  of  the 
axis  can  be  raised  or  lowered. 

(364)  Centring  eye-piece.  In  some  in- 
struments, such  as  that  of  which  a  longitudinal 
section  is  shown  in  the  margin,  the  inner  end 
of  the  eye-piece  may  be  moved  so  that  the 
cross-hairs  shall  be  seen  precisely  in  the  cen- 
tre of  its  field  of  view.  This  is  done  by  means 
of  four  screws,  arranged  in  pairs,  like  those  of 
the  cross-hair-ring  screws,  and  capable  of  mov- 
ing the  eye-piece  up  and  down,  and  to  right 
or  left,  by  loosening  one  and  tightening  the 
opposite  one.  Two  of  them  are  shown  at  A,  A, 
in  the  figure ;  in  which  B,  B,  are  two  of  the 
cross-hair  screws. 

(365)  Centring  object-glass.  In  some 
instruments  four  screws,  similarly  arranged, 
two  of  which  are  shown  at  C,  C,  can  move,  in 
any  direction,  the  inner  end  of  the  slide  which 
carries  the  object-glass.  The  necessity  for 
Buch  an  arrangement  arises  from  the  impossi- 


Fiff.  250. 


a 


rf 


k 


lUjsriJ 


P 


^=S:, 


248        TRANSIT  AND  THEODOLITE  SFRVEYmC.       [part  iv 

bility  of  drawing  a  tube  perfectly  straight.  Consequentlj,  the 
line  of  collimation,  when  the  tube  is  drawn  in,  will  not  coincide  with 
the  same  hne  when  the  tube  is  pushed  out.  If  adjusted  for  one 
position,  it  will  therefore  be  wrong  for  the  other.  These  screws, 
however,  can  make  it  right  m  both  positions.  They  are  used  aa 
follows. 

Sight  to  some  well  defined  point  as  far  off  as  it  can  be  distinctly 
seen.  Then  revolve  the  telescope  half  around  in  its  supports ; 
i.  e.  turn  it  upside  down.*  If  the  line  of  collimation  was  not  in 
the  imaginary  axis  of  the  rings  or  collars  on  which  the  telescope 
rests,  it  will  now  no  longer  bisect  the  object  sighted  to.  Thus, 
if  the  horizontal  hair   was  too  high,  as  in  Fig.  251,  tLis  line  of 

Fig.  251. 

collimation  would  point  at  first  to  A,  and  after  being  turned  over,  it 
would  point  to  B.  The  error  is  doubled  by  the  reversion,  and  it 
should  point  to  C,  midway  between  A  and  B.  Make  it  do  so,  by  un 
screwing  the  upper  capstan-headed  screw,  and  screwing  in  the  lower 
one,  till  the  horizontal  hair  is  brought  half  way  back  to  the  point. 
Remember  that  in  an  erecting  telescope,  the  cross-hairs  are  reversed, 
and  vice  versa.  Bring  it  the  rest  of  the  way  by  means  of  the 
parallel  plate  screws.  Then  revolve  it  in  the  Ys  back  to  its  orig- 
inal position,  and  see  if  the  intersection  of  the  cross-hairs  now 
bisects  the  point,  as  it  should.  If  not,  again  revolve,  and  repeat 
the  operation  tUl  it  is  perfected.  If  the  vertical  hair  passes  to  the 
nght  or  to  the  left  of  the  point  when  the  telescope  is  turned  half 
around,  it  must  be  adjusted  in  the  same  manner  by  the  other  pair 
of  cross-hairs  screws.  One  of  these  adjustments  may  disturb  the 
other,  and  they  should  be  repeated  alternately.  When  they  are 
perfected,  the  intersection  of  the  cross-hairs,  when  once  fixed  on  a 
point,  will  not  move  from  it  when  the  telescope  is  revolved  in  its 

In  Theodolites,  the  Telescope  is  revolved  in  the  Ys.  I»  Transits,  the  maker, 
by  whom  this  adjustment  is  usually  performed,  revolves  the  Telescope,  in  tho 
same  manner,  before  it  is  fixed  in  its  cross-bar. 


CHAP.  HI.]  Adjustments.  249 

supports.  This  double  operation  is  called  adjusting  the  line  of 
collimation.* 

This  line  is  now  adjusted  for  distant  objects.  It  would  be  so  for 
near  ones  also,  if  the  tube  were  perfectly  straight.  To  test  this, 
sight  to  some  point,  as  near  as  is  distinctly  visible.  Then  turn  the 
telescope  half  over.  If  the  intersection  does  not  now  bisect  the 
point,  bring  it  half  way  there  by  the  screws  C,  C,  of  Fig.  250, 
moving  only  one  of  the  hairs  at  a  time,  as  before.  Then  repeat 
the  former  adjustment  on  the  distant  object.  If  this  is  not  quite 
perfect,  repeat  the  operation. 

This  adjustment,  in  instruments  thus  arranged,  should  precede 
the  first  one  which  we  have  explained.  It  is  usually  performed 
by  the  maker,  and  its  screws  are  not  visible  in  the  Transit,  being 
enclosed  in  the  ball  seen  where  the  telescope  is  connected  with 
the  cross-bar.  t 

All  the  adjustments  should  be  meddled  \Nath  as  little  as  possible, 
lest  the  screws  should  get  loose  ;  and  when  once  made  right  they 
should  be  kept  ?o  by  careful  usage. 

*  This  "adjustment  of  the  line  of  collimation"  has  merely  brought  the  intersec- 
rion  of  the  cross-hairs  (which  fixes  the  line  of  sight)  into  the  line  joining  the  cen- 
tres of  the  collars  on  which  the  telescope  turns  in  the  Ys  ;  but  the  maker  is  sup- 
posed to  have  originally  fixed  the  optical  axis  of  thetelescopo  (i.  e.  the  line  joining 
the  optical  centres  "i  the  glasses)   in  the  same  line. 

\  The  adjustment  -f  "  Centring  tlie  (;I)ject-glass  is  the  invention  of  Messrfc 
Gurley,  of  Truy. 


250  fPART  If 


CHAPTER  IV. 


THE  FIELD-WORK. 

(366)  To  measure  a  horizontal  angle.     Set  up  the  instrument 
80  that  its  centre  shall  be  Fig-  252. 

exactly  over  the  angu- 
lar point,  or  in  the  in- 
tersection of  the  two 
lines  whose  difference  of 


AO- 


direction  is  to  be  measured;  as  at  B  in  the  figure.  A  plumb 
line  must  be  suspended  from  under  the  centre.  Dropping  a 
stone  is  an  imperfect  substitute  for  this.  Set  the  instrument 
80  that  its  lower  parallel  plate  may  be  as  nearly  horizontal  as 
possible.  The  levels  will  serve  as  guides,  if  the  four  parallel-plate 
screws  be  first  so  screwed  up  or  down  that  equal  lengths  of  them 
shall  be  above  the  upper  plate.  Then  level  the  instrument  care- 
fully, as  in  Art.  (338).  Direct  the  telescope  to  a  rod,  stake,  or 
other  object,  A  in  the  figure,  on  one  of  the  lines  which  form  the 
angle.  Tighten  the  clamps,  and  by  the  tangent-screw,  (see  Art. 
(336)),  move  the  telescope  so  that  the  intersection  of  the  cross- 
hairs shall  very  precisely  bisect  this  object.  Note  the  reading  of 
the  vernier,  as  explained  in  the  preceding  chapter.  Then  loosen 
the  clamp  of  the  vernier,  and  direct  the  telescope  on  the  other  line 
(as  to  C)  precisely  as  before,  and  again  read.  The  difference  of 
the  two  readings  will  be  the  desired  angle,  ABC.  Thus,  if  the 
first  reading  had  been  40°  and  the  last  190°,  the  angle  would  be 
150°.  If  the  vernier  had  passed  360°  in  turning  to  the  second 
object,  360°  should  be  added  to  the  last  reading  before  subtract- 
ing. Thus,  if  the  first  reading  had  been  300°,  and  the  last  read- 
ing 90°,  the  angle  would  be  found  by  calling  the  last  reading,  as 
it  really  is,  360°  -f  90°  =  450°,  and  then  subtracting  300°. 

It  is  best  to  sight  first  to  the  left  hand  object  and  then  to  the 
right  hand  one,  turning  "  with  the  sun,"  or  like  the  hands  of  a 
watch,  since  the  numbering  of  the  degrees  usually  runs  in  that 
direction. 


CHAP.  IV.]  The  Field-work.  251 

It  is  convenient,  though  not  necessary,  to  begin  by  setting  the 
vernier  at  zero,  by  the  ujDper  movement  (that  of  the  vernier  plate 
on  the  circle)  and  then,  by  means  of  the  lower  motion,  (that  of 
the  whole  instrument  on  its  axis),  to  direct  the  telescope  to  the  first 
object.  Then  fasten  the  lower  clamp,  and  sight  to  the  second 
object  as  before.  The  reading  will  then  be  the  angle  desired. 
An  objection  to  this  is  that  the  two  verniers  seldom  read  alike.* 

After  one  or  more  angles  have  been  observed  from  one  pomt, 
the  telescope  must  be  directed  back  to  the  first  object,  and  the 
reading  to  it  noted,  so  as  to  make  sure  that  it  has  not  slipped. 
A  watch-telescope  (see  Art.  339)  renders  this  unnecessary. 

The  error  arising  from  the  instrument  not  being  set  precisely 
over  the  centre  of  the  station,  will  be  greater  the  nearer  the  object 
sighted  to.  Thus  a  difference  of  one  inch  would  cause  an  error  of 
only  3"  in  the  apparent  direction  of  an  object  a  mile  distant,  but 
one  of  nearly  3'  at  a  distance  of  a  hundred  feet. 

(367)  Reduction  of  liiJ?h  and  low  objects.  AMien  one  of  the 
objects  sighted  to  is  higher  than  the  other,  the  "  plunging  tele- 
scope" of  these  mstruments  causes  the  angle  measured  to  be  the 
true  horizontal  angle  desired ;  i.  e.  the  same  angle  as  if  a  point 
exactly  under  the  high  object  and  on  a  level  with  the  low  object 
(or  vice  versa)  had  been  sighted  to.  For,  the  telescope  has  been 
caused  to  move  in  a  vertical  plane  by  the  3d  adjustment  of  Chap- 
ter II,  and  the  angle  measured  is  therefore  the  angle  between  the 
vertical  planes  which  pass  through  the  two  objects,  and  which 
"  project"  the  two  lines  of  sight  on  the  same  horizontal  plane. 

This  constitutes  the  great  practical  advantage  of  these  instru- 
ments over  those  which  are  held  in  the  planes  of  the  two  objects 
observed,  such  as  the  sextant,  and  the  "  circle"  much  used  by  the 
French. 


*  The  learuer  will  do  well  to  gauge  his  own  precision  and  that  of  the  instrurnent 
(end  he  may  rest  assured  that  his  own  will  be  the  one  chiefly  iu  fault)  by  measur. 
ing,  from  any  station,  the  angles  between  successive  points  all  around  him,  till  he 
gets  back  to  the  first  point,  beginning  at  different  parts  of  the  circle  for  each  an^le. 
The  sum  of  all  these  angles  should  exactly  equal  3()0».  He  will  probably  find 
qaite  a  difference  from  that. 


SDii  TRANSIT  AND  THEODOLITE  SURFEYIXG.    [part  17 

(368)  Notation  of  angles.  The  angles  observed  may  be 
noted  in  various  ways.  Thus,  the  observation  of  the  angle  ABC, 
in  Fig.  252,  may  be  noted  "  At  B,  from  A  to  C,  150°,"  or  better, 
"  At  B,  between  A  and  C,  150°."     In  column  form,  this  becomes 

Between  A  150'|and  C. 
At  B  I 

When  the  vernier  had  been  set  at  zero  before  sighting  to  the 
first  object,  and  other  objects  were  then  sighted  to,  those  objects, 
the  readings  to  which  were  less  than  180°,  will  be  on  the  left  of 
the  first  line,  and  those  to  which  the  readings  were  more  than 
180^,  will  be  on  its  right,  looking  in  the  direction  in  which  the  sur- 
vey is  proceeding,  from  A  to  B,  and  so  on.* 

(369)  Probable  error.  When  a  number  of  separate  observa 
tions  of  an  angle  have  been  made,  the  mean  or  average  of  them  all, 
(obtained  by  dividing  the  sum  of  the  readings  by  their  number,) 
is  taken  as  the  true  reading.  The  "  Probable  error"  of  this  mean, 
is  the  quantity,  (minutes  or  seconds)  which  is  such  that  there  is  an 
even  chance  of  the  real  error  being  more  or  less  than  it.  Thus, 
if  ten  measurements  of  an  angle  gave  a  mean  of  35°  18',  and  it 
was  an  equal  wager  that  the  error  of  this  result,  too  much  or  too 
little,  was  half  a  minute,  then  half  a  minute  would  be  the  "  Probable 
error"  of  this  determination.  This  probable  error  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  errors  (i.  e.  the  differ- 
ences of  each  observation  from  the  mean)  divided  by  the  number 
of  observations,  and  multiplied  by  the  decimal  0.674489. 

The  same  result  would  be  obtained  by  using  what  is  called 
"  The  loeigM''  of  the  observation.  It  is  equal  to  the  square  of 
the  number  of  observations  divided  by  twice  the  sum  of  the  squares 
of  the  errors.  The  "  Probable  error"  is  equal  to  0.476936  divided 
by  the  square  root  of  the  weight.  These  rules  are  proved  by  th© 
"  Theory  of  Probabilities." 

(370)  To  repeat  an  an^le.     Begin  as  m  Art.  (366),  au 
measure  the   angle   as   there   directed.      Then  unclamp  below, 
and  turn  the  circle  around  till  the  telescope  is  again  directed  to 
the  first  object,  and  made  to  bisect  it  precisely  by  the  lower  tan- 

'This  is  very  useful  in  preventing  any  ambiguity  in  the  field-notes 


CHAP.  IT.]  The  Field-work.  253 

gent  screw.  Then  unclamp  above  and  turn  the  vernier  plate  till 
the  telescope  again  points  to  the  second  object,  the  first  reading 
remaining  unchanged.  The  angle  will  now  have  been  measured  a 
second  time,  but  on  a  part  of  the  circle  adjoining  that  on  which  it 
was  first  measured,  the  second  arc  beginning  where  the  first  ended. 
The  difference  between  the  first  and  last  readings  will  therefore  be 
twice  the  angle. 

This  operation  may  be  repeated  a  third,  a  fourth,  or  any  num- 
ber of  times,  always  turning  the  telescope  back  to  the  first  object 
by  the  lower  movement,  (so  as  to  start  with  the  reading  at  which 
the  preceding  observation  left  ofi")  and  turning  it  to  the  second 
object  by  the  upper  movement.  Take  the  difibrence  of  the  first 
and  last  readings  and  divide  by  the  number  of  observations. 

The  advantage  of  this  method  is  that  the  errors  of  observation 
(i.  e.  sighting  sometimes  to  the  right  and  sometimes  to  the  left  of 
the  true  point)  balance  each  other  in  a  number  of  repetitions  ; 
while  the  constant  error  of  graduation  is  reduced  in  proportion  to 
this  number.  This  beautiful  prin^-iple  has  some  imperfections  in 
practice,  probably  arising  from  the  slipping  and  straining  of  the 
clamps. 

(371)  Angles  of  deflection.  The  angle  of  deflection  of  one 

line  from  another,  is  the  ^'ig-  253. 

angle    which    one    line  ^'^ 

makes   with    the    other  ^y'^ 

line  produced.     Thus, in  /f^^%c  «■ 

AG ^ — (H^J 

the  figure,  the  angle  of  \=-/ 

deflection   of   BC    from 

AB,  is  B'BC.     It  is  evidently  the  supplement  of  the  angle  ABC 

To  measure  it  with  the  Transit,  set  the  instrument  at  B,  direct 
the  telescope  to  A,  and  then  turn  it  over.  It  will  now  pomt  in  the 
direction  of  AB  produced,  or  to  B',  if  the  2d  adjustment  of  Chapter 
EL,  has  been  performed.  Note  the  reading.  Then  dii'ect  the 
telescope  to  C.  Note  the  new  reading,  and  their  difference  will 
be  the  required  angle  of  deflection,  B'BC. 

If  the  vernier  be  set  at  zero,  before  taking  the  first  observation, 
the  readings  for  objects  on  the  right  of  the  first  line  ^-ill  be  less  than 


254        TRANSIT  Ai\D  THEODOLITE  SURVEYING.       [part  it 

180°,  and  more  than  180"  for  objects  on  the  left ;  conversely  to 
Art.  (368). 

(372)  Line  surveying.  The  survey  of  a  line,  such  as  a  road, 
&c.,  can  be  made  by  the  Theodolite  or  Transit,  with  great  precis- 
ion ;  measuring  the  angle  which  each  line  makes  with  the  preced- 
ing line,  and  noting  their  lengths,  and  the  necessary  offsets  on  each 
side. 

Short  lines  of  sight  should  be  avoided,  since  a  slight  inaccuracy 
in  setting  the  centre  of  the  instrument  exactly  over  or  under  the 
point  previously  sighted  to,  would  then  much  affect  the  angle,  as 
noticed  at  close  of  Art.  (366).  Very  great  accuracy  can  be  ob- 
tained by  using  three  tripods.  One  would  be  set  at  the  first  sta- 
tion and  sighted  back  to  from  the  instrument  placed  at  the  second 
station,  and  a  forward  sight  be  then  taken  to  the  third  tripod  placed 
at  the  third  station.  The  instrument  would  then  be  set  on  this 
third  tripod,  a  back  sight  taken  to  the  tripod  remaining  on  the  se- 
cond station,  and  a  foresight  taken  to  the  tripod  brought  from  the 
first  station  to  the  fourth  station ;  to  which  the  instrument  is  next 
taken :  and  so  on.     This  is  especially  valuable  in  surveys  of  mines. 

The  field-notes  may  be  taken  as  directed  in  Chapter  III  of  Com- 
pass Surveying,  pages  149,  &c.,  the  angles  taking  the  place  of 
the  Bearings.  The  "  Checks  by  intersecting  Bearings,"  explained 
in  Art.  (246),  should  also  be  employed.  The  angles  made  on  each 
side  of  the  stations  may  both  be  measured,  and  the  equahty  of  their 
sum  to  360°,  would  at  once  prove  the  accuracy  of  the  work. 

If  the  magnetic  Bearing  of  any  one  of  the  lines  be  given,  and 
that  of  any  of  the  other  lines  of  the  series  be  required,  it  can  be 
deduced  by  constructing  a  diagram,  or  by  modifications  of  the  rules 
given  for  the  reverse  object,  in  Art.  (213). 

(373)  TraTcrsing  :  Or  Surveying  by  the  back-angle.    This  is 

a  method  of  observing  and  recording  the  different  directions  of  suc- 
cessive portions  of  a  line,  (such  as  a  road,  the  boundaries  of  a  farm, 
&c.,)  so  as  to  read  off  on  the  instrument,  at  each  station,  the  angle 
which  each  line  makes — not  with  the  preceding  line,  but — with  the 
first  hue  observed.  This  line  is,  therefore,  called  the  meridian  of 
that  survey. 


«HAP.  IV.] 


The  Field-work. 

Fig.  254. 


255 


Set  up  the  instrument  at  the  first  angle,  or  second  station,  (B, 
in  the  figure),  of  the  line  to  be  surveyed.  Sight  to  A  and  then  to 
C.  Clamp  the  vernier,  and  take  the  instrument  to  C.  Loosen 
the  lower  clamp,  and  direct  the  telescope  to  B,  the  reading  remain- 
ing as  it  was  at  B.  Clamp  below,  loosen  above,  and  sight  to  D. 
The  reading  of  the  instrument  will  be  the  angle  which  the  line  CD 
makes  with  the  first-  line,  or  Meridian,  AB. 

Take  the  instrument  to  D.  Sight  back  to  C,  and  then  forward 
to  E,  as  before  directed,  and  the  reading  of  the  instrument  will  be 
the  angle  which  DE  makes  with  AB. 

So  proceed  for  any  number  of  lines. 

When  the  Transit  is  used,  the  angles  of  deflection  of  each  line 
from  the  first,  obtained  by  reversing  the  telescope,  may  be  used  in 
"  Traversing,"  and  with  much  advantage  when  the  successive 
lines  do  not  differ  greatly  in  their  directions. 


The  survey  represented  in  the  figure, 
is  recorded  in  the  first  of  the  accompa- 
nying Tables,  as  observed  with  the  The- 
odolite ;  and  in  the  second  Table,  as 
observed  with  the  Transit. 


The  chief  advantage  of  this  method  is  its  greater  rapidity  in  the 
field  and  in  platting,  the  angles  being  all  laid  down  from  one  meri- 
dian, as  in  Compass-surveying.  This  also  increases  the  accuracy 
of  the  plat,  since  any  error  in  the  direction  of  one  line  does  not 
affect  the  directions  of  the  following  lines.* 

(374)  Use  of  the  Compass.  The  chief  use  of  the  Compass 
attached  to  a  Transit  or  Theodohte,  is  as  a  check  on  the  observa- 
tions ;    for  the  difference  between  the  magnetic  Bearings  of  any 

•  If  there  are  two  verniers ;  take  care  always  to  read  the  degrees  from  the 
same  vernier.     Mark  it  A. 


A 

0° 

B 

200° 

C 

50° 

D 

ISO" 

E 

300" 

F 

210" 

G 

250" 

A 

0' 

B 

20° 

C 

50° 

D 

0^ 

E 

300° 

F 

30° 

G 

250° 

255       TRANSIT  AXD  THEODOLITE  SURVEYING,      [part  iv. 

two  lines  should  be  the  same,  approximately,  as  the  angle  between 
them,  measured  by  the  more  accurate  instruments.  The  Bearing 
also  prevents  any  ambiguity,  as  to  whether  an  angle  was  taken  to 
the  right  or  to  the  left. 

The  instrument  may  also  be  used  like  a  simple  compass,  the  tele- 
scope taking  the  place  of  the  sights,  and  requiring  similar  tests  of 
accuracy.  A  more  precise  way  of  taking  a  Bearing  is  to  turn  the 
plate  to  which  the  compass  box  is  attached,  tiU  the  needle  points 
to  zero,  and  note  the  reading  of  the  vernier ;  then  sight  to  the 
object,  and  again  read  the  vernier.  The  Bearing  will  thus  be 
obtained  more  minutely  than  the  divisions  on  the  compass  box 
could  give  it. 

(375)  Measuring  distances  with  a  telescope  and  rod.     On 

the  cross-hair  ring,  described  in  Art.  (330),  stretch  two  more  hori- 
zontal spider-threads  at  equal  distances  above  and  below  the  origi- 
nal one  ;  or  all  may  be  replaced  by  a  plate  of  thin 
glass,  placed  precisely  in  the  focus,  with  the  necessary 
lines,  as  in  the  figure,  etched  by  fluoric  acid.  Let  a 
rod,  10  or  15  feet  long,  be  held  up  at  1000  feet  off,  and 
let  there  be  marked  on  it  precisely  the  length  which 
the  distance  between  two  of  these  lines  covers.  Let  this  be  subdi- 
vided as  minutely  as  the  spaces,  painted  alternately  white  and 
red  and  numbered,  can  be  seen.  If  ten  subdivisions  are  made, 
each  will  represent  a  distance  of  100  feet  off,  and  so  on.  Con- 
tinue these  divisions  over  the  whole  length  of  the  rod.  It  is  now 
ready  for  use.  The  French  call  it  a  stadia.  When  it  is  held  up 
at  any  unknown  distance,  the  number  of  divisions  on  it  intercepted 
between  the  two  lines,  will  indicate  the  distance  with  considerable 
precision.    It  should  be  tested  at  various  distances. 

A  "  Levelling-rod,"  divided  into  feet,  tenths  and  hundredths, 
may  be  used  as  a  stadia,  with  less  convenience  but  more  precision. 
Experiments  must  previously  determine  at  what  distances  the 
space  between  the  lines  in  the  telescope  covers  one  foot,  &c.  Then, 
at  any  unknown  distance,  let  the  sUding  "  target"  of  the  rod  be 
moved  till  one  line  bisects  it,  and  its  place  on  the  rod  be  read  off; 
let  the  target  be  then  moved  so  that  the  other  Ime  bisects  it  and  let 


Fig. 

255. 

\ 

V 

/ 

CHAP.  IV.]  The  Field-work.  257 

its  place  be  again  noted.  Then  the  required  distance  will  be  equal  to 
the  difference  of  the  readings  on  the  rod,  in  feet,  multiplied  by  the 
distance  at  which  a  foot  was  intercepted  between  the  lines. 

One  of  the  horizontal  hairs  may  be  made  movable,  and  its  dis- 
tance from  the  other,  when  the  space  between  them  exactly  covers 
an  object  of  known  height,  can  be  very  precisely  measured  by 
counting  the  number  of  turns  and  fractions  of  a  turn,  of  a  screw 
by  which  this  movable  hair  is  raised  or  lowered.  A  simple  pro- 
portion will  then  give  the  distance. 

On  sloping  ground  a  double  correction  is  necessary  to  reduce 
the  slope  to  the  horizon  and  to  correct  the  oblique  view  of  the  rod. 
The  horizontal  distance  is,  in  consequence,  approximately  equal  to 
the  observed  distance  multiplied  by  the  square  of  the  cosine  of  the 
slope  of  the  ground. 

The  latter  of  the  above  two  corrections  .wUl  be  dispensed  with 
by  holding  the  rod  perpendicular  to  the  Une  of  sight,  with  the  aid 
of  a  right  angled  triangle,  one  side  of  which  coincides  with  the  rod 
at  the  height  of  the  telescope,  and  the  other  side  of  which  adjoining 
the  right  angle,  is  caused,  by  leaning  the  rod,  to  point  to  the  tele- 
scope. 

Other  contrivances  have  been  used  for  the  same  object,  such  as 
a  Binocular  Telescope  with  two  eye-pieces  inclined  at  a  certain 
angle ;  a  Telescope  with  an  object-glass  cut  into  two  movable 
parts;  &c.  -ty-cA) 

(376)  Ranging-  ont  lines.  This  is  the  converse  of  Survevmg 
lines.  The  instrument  is  fixed  over  the  first  station  with  great 
precision,  its  telescope  being  very  carefully  adjusted  to  move  in  a 
vertical  plane.  A  series  of*  stakes,  with  nails  driven  in  their  tops, 
or  otherwise  well  defined,  are  then  set  in  the  desired  hne  as  far 
as  the  power  of  the  instrument  extends.  It  is  then  taken  forward 
to  a  stake  three  or  four  from  the  last  one  set,  and  is  fixed  over  it, 
first  by  the  plumb  and  then  by  sighting  backward  and  forward  to 
the  first  and  last  stake.  The  line  is  then  continued  as  before.  A 
good  object  for  a  long  sight  is  a  board  painted  hke  a  target,  with 
black  and  white  concentric  rings,  and  made  to  slide  in  grooves  cut 
m  the  tops  of  two  stakes  set  in  the  ground  about  in  tlie  line.     It 

17 


258  TRANSIT  AXD  TDEODOLITE  SURVEYING,    [part  17 

IS  moved  till  the  vertical  hair  bisects  the  circles  (vrhich  the  eye 
can  determine  with  great  precision)  and  a  plumb-line  dropped  from 
their  centre,  gives  the  place  of  the  stake.  "  Mason  &  Dixon's 
Line"  was  thus  ranged. 

If  a  Transit  be  used  for  ranging,  its  "  Second  Adjustment"  is 
most  important  to  ensure  the  accuracy  of  the  reversal  of  its  Tele- 
scope. If  a  Theodolite  be  used,  the  line  is  continued  by  turning 
the  vernier  180°,  or  by  reversing  the  telescope  in  its  Ys,  as  noticed 
in  Arts.  (325)  and  (362). 

(377)  Farm  Surveying,  &c,  A  large  farm  can  be  most  easily 
and  accurately  surveyed,  by  measuring  the  angles  of  its  main  boun- 
daries (and  a  few  main  diagonals,  if  it  be  very  large,)  with  a  The- 
odolite or  Transit,  as  in  Arts.  (866)  or  (371),  and  filling  up  the 
interior  details,  as  fences,  &c.,  with  the  Compass  and  Chain. 

If  the  Theodolite  be  used.  Fig.  25G. 

keep  the  field  on  the  left 
hand,  as  in  following  the  or- 
der of  the  letters  in  this 
figure,  and  turn  the  telescope 
around  "  with  the  sun,"  and 
the  Lngles  measured  as  in  d< 
Art.  (366),  will  be  the  interior  angles  of  the  field,  as  noted  m  the 
figure. 

The  accuracy  of  the  work  will  be  proved,  as  alluded  to  in 
Art.  (257),  if  the  sum  of  all  the  interior  angles  be  equal  to  the  pro- 
duct of  180°  by  the  number  of  sides  of  the  figure  less  two.  Thua 
in  the  figure,  the  sum  of  all  the  interior  angles  =  540°  =  180°  X 
(5  —  2).  The  sum  of  the  exterior  angles  would  of  course  equal 
180°  X  (5  -f-  2)  =  1260°. 

If  th«  Transit  be  used,  the  farm  should  be  kept  on  the  right 
hand,  and  then  the  angles  measured  will  be  the  supplements  of  the 
mterior  angles.  If  the  angles  to  tlie  right  be  called  j^ositive,  and 
those  to  the  left  negative,  their  algebraic  sum  should  equal  360°. 

If  the  boundary  lines  be  surveyed  by  "  Traversing,"  as  in  Art. 
(373),  the  reading,  on  getting  back  to  the  last  station  and  looking 
back  to  the  first  line,  should  be  360°,  or  0°. 


CHAP.  IV.]  The  Field-work.  259 

The  content  of  any  surface  surveyed  by  "  Traversing  "  with  the 
Transit  can  be  calculated  by  the  Traverse  Table,  as  in  Chapter 
"\T[,  of  Part  III,  by  the  following  modification.  A\Tien  the  angle 
of  deflection  of  any  side  from  the  first  side,  or  Meridian,  is  less  than 
90'',  call  this  angle  the  Bearing,  find  its  Latitude  and  Departure, 
and  call  them  both  plus.  When  the  angle  is  between  90°  and 
180°,  call  the  difference  between  the  angle  and  180°  the  Bearuig, 
and  caU  its  Latitude  minus  and  its  Departure  plus.  When  the 
angle  is  between  180°  and  270°,  call  its  difference  from  180°  the 
Bearing,  and  caU  its  Latitude  minus  and  its  Departure  minus. 
"When  the  angle  is  more  than  270°,  call  its  difference  from  360° 
the  Bearing,  and  call  its  Latitude  plus  and  its  Departure  minus. 
Then  use  these  as  in  getting  the  content  of  a  Compass-survey. 
The  signs  of  the  Latitudes  and  Departures  foUow  those  of  the 
cosmos  and  sines  in  the  successive  quadrants. 

Town-Surveying  would  be  performed  as  directed  in  Art.  (261), 
substituting  "  angles"  for  "  Bearings."  "  Traversing"  is  the  best 
method  in  all  these  cases. 

Inaccessible  areas  would  be  surveyed  nearly  as  in  Art.  (134), 
except  that  the  angles  of  the  hnes  enclosing  the  space  would  be 
measured  with  the  instrument,  instead  of  with  the  chaui. 

(378)  Platting.  Any  of  these  surveys  can  be  platted  by  any 
of  the  methods  explained  and  charactemed  in  Chapter  TV,  of  the 
preceding  Part.  A  circular  Protractor,  Art.  (264),  may  be 
regarded  as  a  Theodolite  placed  on  the  paper.  "  Plattmg  Bear- 
ings," Art.  (265),  can  be  employed  when  the  survey  has  been 
made  by  "  Traversing."  But  the  method  of  "  Latitudes  and  De- 
partures," Art.  (285),  is  by  far  the  most  accurate. 


PAET  V 


TRIANGULAR  SURVEYING; 

OR 

By  Hie  Fourth  Method. 

(379)  Triangular  Surveying  is  founded  on  the  Fourth  Metlwa 
of  determining  the  position  of  a  point,  by  the  intersection  of  two 
known  lines,  as  given  in  Art.  (8).  By  an  extension  of  the  princi* 
pie,  a  field,  a  farm,  or  a  country,  can  be  surveyed  by  measuring 
only  one  line,  and  calculating  all  the  other  desired  distances,  which  are 
made  sides  of  a  connected  series  of  imaginary  Triangles,  whose 
)  angles  are  carefully  measured.     The  district  surveyed  is  covered 

-i  with  a  sort  of  net-work  of  such  triangles,  whence  the  name  given  to 

\l  this  kind  of  Surveying.     It  is  more  commonly  called  "  Trigonome- 

trical Surveying;"  and  sometimes  "  Geodesic  Surveying,"  but  im- 
properly, since  it  does  not  necessarily  take  into  account  the  curv- 
ature of  the  earth,  though  always  adopted  in  the  great  surveys  in 
which  that  is  considered. 

(380)  Outline  of  operations.     A  base  line,  as  long  as  possible, 

(5  or  10  miles  in  surveys  of  countries),  is  measured  with  extreme 
accuracy. 

From  its  extremities,  angles  are  taken  to  the  most  distant  objects 
visible,  such  as  steeples,  signals  on  mountain  tops,  &c. 

The  distances  to  these  and  between  these  are  then  calculated  by 
the  rules  of  Trigonometry. 

The  mstrument  is  then  placed  at  each  of  these  new  stations,  and 
angles  are  taken  from  them  to  still  more  distant  stations,  the  calcu- 
lated lines  being  used  as  new  base  lines. 

This  process  is  repeated  and  extended  till  the  whole  district  is 
embraced  by  these  "  primary  triangles"  of  as  large  sides  as  possible. 


PARTY  J  TRIANGULAR  SURVEYING.  261 

One  side  of  the  last  triangle  is  so  located  that  its  length  can  be 
obtained  bj  measurement  as  well  as  bj  calculation,  and  the  agree- 
ment of  the  two  proves  the  accuracy  of  the  whole  work. 

Within  these  primary  triangles,  secondary  or  smaller  triangles 
are  formed,  to  fix  the  position  of  the  minor  local  details,  and  to 
serve  as  starting  points  for  common  surveys  with  chain  and  com- 
pass, &c.     Tertiary  triangles  may  also  be  required. 

The  larger  triangles  are  first  formed,  and  the  smaller  ones  based 
on  them,  in  accordance  with  the  important  principle  in  all  survey- 
mg  operations,  always  to  work  from  the  whole  to  the  parts,  and  from 
greater  to  less. 

Each  of  these  steps  will  now  be  ccnsidered  in  turn,  in  the 
following  order : 

1.  The  Base;  articles  (381),  (382). 

2.  The  Triangulation ;  articles  (383)  to  (390). 

3.  Modifications  of  the  method;  articles  (391)  to  (395). 

(381)  Measuring  a  Base.  Extreme  accuracy  in  this  is  neces- 
sary, because  any  error  in  it  will  be  multiplied  in  the  subsequent 
work.  The  ground  on  which  it  is  located  must  be  smooth  and  nearly 
level,  and  its  extremities  must  be  in  sight  of  the  chief  points  in  the 
neighborhood.  Its  point  of  beginnmg  must  be  marked  by  a  stone 
set  in  the  ground  with  a  bolt  let  into  it.  Over  this  a  TheodoUte 
or  Transit  is  to  be  set,  and  the  line  "ranged  out"  as  directed  in 
Art.  (376).  The  measurement  may  be  made  with  chains,  (which 
should  be  formed  like  that  of  a  watch,)  &c.  but  best  with  rods.  We 
will  notice  in  turn  their  Materials,  Siqjports,  Alinement,  Levelling, 
and  Contact. 

As  to  Materials,  iron,  brass  and  other  metals  have  been  used, 
but  are  greatly  lengthened  and  shortened  by  changes  of  tempera- 
ture. Wood  is  affected  by  moisture.  Glass  rods  and  tubes  are 
preferable  on  both  these  accounts.  But  wood  is  the  most  conve- 
nient. Wooden  rods  should  be  straight-grained  white  pine,  &c. ; 
well  seasoned,  baked,  soaked  in  boihng  oil,  painted  and  varnished. 
They  may  be  trussed,  or  framed  like  a  mason's  plumb-hne  level,  to 
prevent  their  bending.  Ten  or  fifteen  feet  is  a  convenient  length. 
Three  are  required,  which  may  be  of  different  colors,  to  prevent 


262  TRIAIVGULAR  SURVEYLXG.  [part  ▼ 

mistakes  in  recording.  They  must  be  very  carefully  compared 
Tvith  a  standard  measure. 

Supports  must  be  provided  for  the  rods,  in  accurate  -R-ork. 
Posts  set  in  line  at  distances  equal  to  the  length  of  the  rods,  may 
be  driven  or  sawed  to  a  uniform  line,  and  the  rods  laifi  on  them, 
either  directly,  or  on  beams  a  little  shorter.  Tripods,  or  trestles, 
with  screws  in  their  tops  to  raise  or  lower  the  ends  of  the  rods 
resting  on  them,  or  blocks  with  three  long  screws  passing  through 
them  and  serving  as  legs,  may  also  be  usbd.  Staves,  or  legs,  for 
the  rods  have  been  used ;  these  legs  bearing  pieces  which  can  shde 
up  and  down  them  and  on  which  the  rods  themselves  rest. 

The  AUnement  of  the  rods  can  be  effected,  if  they  are  laid  on 
the  ground,  by  strings,  two  or  three  hundred  feet  long,  stretched 
between  the  stakes  set  in  the  line,  a  notched  peg  being  driven  when 
the  measurement  has  reached  the  end  of  one  string,  which  is  then 
taken  on  to  the  next  pair  of  stakes  ;  or,  if  the  rods  rest  on  supports, 
by  projecting  points  on  the  rods  being  ahned  by  the  instrument. 

The  Levelling  of  the  rods  can  be  performed  with  a  common 
mason's  level ;  or  their  angle  measured,  if  not  horizontal,  by  a 
"  slope-level." 

The  Contacts  of  the  rods  may  be  effected  by  bringing  them  end 
to  end.  The  third  rod  must  be  applied  to  the  second  before  the 
first  has  been  removed,  to  detect  any  movement.  The  ends  must 
be  protected  by  metal,  and  should  be  rounded  (with  radius  equal 
to  length  of  rod)  so  as  to  touch  in  only  one  point.  Round-headed 
nails  will  answer  tolerably.  Better  are  small  steel  cyHnders,  hori- 
zontal on  one  end  and  vertical  on  the  other.  Shding  ends,  with 
verniers,  have  been  used.  If  one  rod  be  higher  than  the  next  one, 
one  must  be  brought  to  touch  a  plumb-line  which  touches  the  other, 
and  its  thickness  be  added.  To  prevent  a  shock  from  contact,  the 
rods  may  be  brought  not  quite  in  contact,  and  a  wedge  be  let  down 
between  them  till  it  touches  both  at  known  points  on  its  graduated 
edges.  The  rods  may  be  laid  side  by  side,  and  lines  drawn  across 
the  end  of  each  be  made  to  coincide  or  form  one  line.  This  is  more 
accurate.  Still  better  is  a  "  visual  contact,"  a  double  microscope 
with  cross-hairs  being  used,  so  placed  that  one  tube  bisects  a  dot 
at  the  end  of  one  rod,  and  the  other  tube  bisects  a  dot  at  the  end 


PART  v.]  TRIANGULAR  SCRVEYIXG.  2G3 

of  the  next  rod.    The  rods  thus  never  touch.     The  distance 
between  the  two  sets  of  cross-hairs  is  of  course  to  be  added. 

A  Base  could  be  measured  over  very  uneven  ground,  or  even 
Avater,  by  suspendmg  a  series  of  rods  from  a  stretched  rope  by 
rings  in  which  they  can  move,  and  leveUing  them  and  bringing 
them  into  contact  as  above. 

(382)  Corrections  of  Base.  If  the  rods  were  not  level,  their 
length  must  be  reduced  to  its  horizontal  projection.  This  wDuld 
be  the  square  root  of  the  diflference  of  the  squares  of  the  length  of 
the  rod  (or  of  the  base)  and  of  the  height  of  one  end  above  the 
5ther ;  or  the  product  of  the  same  length  by  the  cosine  of  the 
angle  which  it  makes  with  the  horizon.* 

If  the  rods  were  metallic,  they  would  need  to  be  ccrrected  for 
temperature.  Thus,  if  an  iron  bar  expands  roooooo  of  its  length 
for  1°  Fahrenheit,  and  had  been  tested  at  32°,  and  a  Base  had  been 
measured  at  72°  with  such  a  bar  10  feet  long,  and  found  to  contain 
3000  of  them,  its  apparent  length  would  be  30,000  feet,  but  iti 
real  length  would  be  SA  feet  more.  An  iron  and  a  brass  ba 
can  be  so  combined  that  the  difference'  of  their  expansion, 
causes  two  points  attached  to  their  ends  to  remain  at  the  sami 
distance  at  all  temperatures.  Such  a  combination  is  used  on 
the  U.  S.  Coast  Survey. 

(383)  Choice  of  Stations.  The  stations,  or  "  Trigonometrical 
points,"  which  are  to  form  the  vertices  of  the  triangles,  and  to  be 
observed  to  and  from,  must  be  so  selected  that  the  resulting  trian- 
gles may  be  "  well-conditioned,"  i.  e.  may  have  such  sides  and  angles 
mat  a  small  error  in  any  of  the  measured  quantities  will  cause  the 
least  possible  errors  in  the  quantities  calculated  from  them.  The 
higher  Calculus  shows  that  the  triangles  should  be  as  nearly  equi- 
lateral as  possible.  This  is  seldom  attainable,  but  no  angle  should 
be  admitted  less  than  30°,  or  more  than  120°. f 

*  Moi-e  precisely,  A  being  this  angle,  and  not  more  than  2°  or  3^*,  the  differ 
ence  between  the  inclined  and  horizontal  lengths,  equals  the  inclined  or  real 
length  multiplied  by  the  square  of  the  minutes  in  A,  and  that  by  the  decimal 
0.00000004231  ;  as  shewn  in  Appendix  B.  In  a  Geodesic  survey,  the  base  would 
also  be  required  to  be  reduced  to  the  level  of  the  sea. 

t  When  two  angles  only  are  observed,  as  is  often  the  case  m  the  secondary 
triargulation,  the  unobserved  angle  ought  to  be  nearly  a  right  angle. 


264 


TRIAXGULAR  SURVEIOG. 


[part  v. 


To  extend  the  triangulation,  by  continually  increasing  the  sides 
of  the  triangles,  without  introducing  "ill-conditioned"  triangles, 
may  be  effected   as  in  the  figure.     AB  is  the  measured  base 


C  and  D  are  the  nearest  stations.  In  the  triangles  ABC  and  ABD, 
all  the  angles  being  observed  and  the  side  AB  known,  the  other 
Bides  can  be  readily  calculated.  Then  in  each  of  the  triangles 
DAC  and  DBC,  two  sides  and  the  contained  angles  are  given  to  find 
DC,  one  calculation  checking  the  other.  DC  then  becomes  a  base 
to  calculate  EF ;  which  is  then  used  to  find  GH  ;  and  so  on. 

The  fewer  primary  stations  used,  the  better ;  both  to  prevent 
confusion  and  because  the  smaller  number  of  triangles  makes  the 
correctness  of  the  results  more  "probable." 

The  United  States  Coast  Survey,  under  the  superintendence  of 
Prof.  A.  D.  Bache,  displays  some  fine  illustrations  of  these  pruici- 
ples,  and  of  the  modifications  they  may  undergo  to  suit  various 
localities.  The  figure  on  the  opposite  page  represents  part  of  the 
scheme  of  the  primary  triangulation  resting  on  the  Massa- 
chusetts base  and  including  some  remarkably  well-conditioned 
triangles,  as  well  as  the  system  of  quadrilaterals  which  is  a  valuable 
feature  of  the  scheme  when  the  sides  of  the  triangles  are  extended 
to  considerable  lengths,  and  quadrilaterals,  with  both  diagonals 
determined,  take  the  place  of  simple  triangles. 

The  engraving  is  on  a  scale  of  1 :  1200,000. 


PART  y 


TRIANGULAR  SURVEYING. 


235 


Fis:.  258 


266 


TRIANGULAR  SURVEYING. 


[part  y 


(384)  Signals.     They  must  be  high,  conspicuous,  and  so  made 
that  the  instrument  can  be  placed  precisely  under  them. 

Three  or  four  timbers  framed   into  a  Fig-  259. 

pyramid,  as  in  the  figure,  with  a  long  mast ' 
projecting  above,  fulfil  the  first  and  last 
conditions.  The  mast  may  be  made  verti- 
cal by  directing  two  theodolites  to  it  and  ad- 
justing it  so  that  their  telescopes  follow  it 
up  and  down,  their  lines  of  sight  being  at 
right  angles  to  each  other.  Guy  ropes 
may  be  used  to  keep  it  vertical. 

A  very  excellent  signal,  used  on  the  Massachusetts  State  Survey, 
by  Mr.  Borden,  is  represented  in  the  three  following  figures.     It 


Fig.  2C0. 


Fig.  261. 


Fig.  2fi2 


consists  merely  of  three  stout  sticks,  which  form  a  tripod,  framed 
with  the  signal  staff,  by  a  bolt  passing  through  their  ends  and  its 
middle.  Fig.  260  represents  the  signal  as  framed  on  the  ground ; 
Fig,  261  shews  it  erected  and  ready  for  observation,  its  base  being 
steadied  with  stones ;  and  Fig.  262  shews  it  with  the  staff  turned 
aside,  to  make  room  for  the  Theodolite  and  its  pro-  Fig.  263 
tecting  tent.  The  heights  of  these  signals  varied  be- 
tween 15  and  80  feet. 

Another  good  signal  consists  of  a  stout  post  let  into 
the  ground,  with  a  mast  fastened  to  it  by  a  bolt  below 
and  a  collar  above.  By  opening  the  collar,  the  mast 
can  be  turned  down  and  the  Theodolite  set  exactly 
under  the  former  summit  of  the  signal,  i.  e.  in  its  verti-  '-  \  )  % 
f'dX  axis.  I,-- ..i»J'..4| 

Signals  should  have  a  height  equal  to  at  least  =^o-oo  of  their  di* 


PART  v.] 


TRIANGULAR  SFRVEYIIVG. 


267 


tance,  sc  as  to  subtend  an  angle  of  half  a  minute,  -which  expe- 
rience has  shown  to  be  the  least  allowable. 

To  make  the  tops  of  the  signal-masts  conspicuous,  flags  may  be 
attached  to  them ;  white  and  red,  if  to  be  seen  against  the  ground, 
and  red  and  green  if  to  be  seen  against  the  sky.*  The  motion  of 
flags  renders  them  visible,  when  mnch  larger  motionless  objects 
are  not.  But  they  are  useless  in  calm  weather.  A  disc  of  sheet- 
iron,  with  a  hole  in  it,  is  very  conspicuous.  It  should  be  arranged 
80  as  to  be  turned  to  face  each  station.  A  barrel,  formed  of  mus- 
lin sewn  together  four  or  five  feet  long,  with  two  hoops  in  it  two 
feet  apart,  and  its  loose  ends  sewn  to  the  signal-staff,  which  passes 
through  it,  is  a  cheap  and  good  arrangement.  A  tuft  of  pine  boughs 
fastened  to  the  top  of  the  staff,  will  be  well  seen  against  the  sky. 

In  sunshine,  a  number  of  pieces  of  tin  nailed  to  the  staff  at  dif- 
ferent angles,  will  be  very  conspicuous.  A  truncated  cone  of 
burnished  tin  will  reflect  the  sun's  rays  to  the  eye  in  almost  every 
situation.  But  a  "  heliotrope,"  which  is  a  piece  of  looking-glass, 
so  adjusted  as  to  reflect  the  sun  directly  to  any  desired  point,  is 
the  most  perfect  arrangement. 

For  night  signals,  an  Argand  lamp  is  used ;  or,  best  of  all.  Drum- 
mond's  light,  produced  by  a  stream  of  oxygen  gas  directed  through 
a  flame  of  alcohol  upon  a  ball  of  lime.  Its  distmctness  is  exceed- 
ingly increased  by  a  parabolic  reflector  behind  it,  or  a  lens  in  front 
of  it.     Such  a  Hght  was  brilliantly  visible  at  66  miles  distance. 


(385)  Observations  of  the  Angles,  These  should  be  repeated 
as  often  as  possible.  In  extended  surveys,  three  sets,  of  ten  each., 
are  recommended.  They  should  be  taken  on  different  parts  of  the 
circle.  In  ordinary  surveys,  it  is  well  to  employ  the  method  of 
"  Traversing,"  Art.  (373).     In  long  sights,  the  state  of  the  atmos- 


*  To  deteraiiiie  at  a  station  A, 
whether  its  signal  can  be  seen 
from  B,  projected  against  the 
eky  or  not,  measure  the  vertical 
angles  BAZ  and  ZAC.  If  their 
»um  equals  or  exceeds  180°,  A 
will  be  thus  seen  from  B.  It 
not,  the  signal  at  A  must  be  rais 
ed  till  this  sum  eciuals  180°. 


268 


TRIWGULAR  SURVEYING. 


[PAET  ▼, 


phere  has  a  very  remarkable  efifect  on  both  the  visibility  of  the 
signals,  and  on  the  correctness  of  the  observations. 

When  many  angles  are  taken  from  one  station,  it  is  important  to 
record  them  by  some  uniform  system.  The  form  given  below  ia 
convenient.  It  will  be  noticed  that  only  the  minutes  and  seconds 
of  the  second  vernier  are  employed,  the  degrees  being  all  taken 
from  the  first. 


Gbservations 

t/ 

• 

1 

STATION 

READINGS. 

MEAN 
READING. 

RIGHT  OR  LEFT  Of 

preced'g  ob.j't. 

REMARKS. 

OBSERVED   TO 

VERNIER   A. 

VERNIER   B. 

A 

B 

C 

70°    19'       0" 

103°  32'  20" 
115°  14'  20" 

18'    40" 

32'  40" 
14'  50" 

70°    18'    50" 

103°  32'  30" 
11.5°  14'  3.5" 

R. 

R. 

When  the  angles  are  "repeated,"  Art.  (370),  the  multiple 
arcs  will  be  registered  under  each  other,  and  the  mean  of  the 
seconds  shewn  by  all  the  verniers  at  the  first  and  last  readings  be 
adopted. 

(386)  Reduction  to  the  centret  It  is  often  impossible  to  set 
the  mstrument  precisely  at  or  under  the  signal  which  has  been 
observed.     In  such  cases  pro-  Fig.  265. 

ceed  thus.  Let  C  be  the  cen^ 
tre  of  the  signal,  and  RCL  the 
desired  angle,  R  being  the  right 
hand  object  and  L  the  left  hand 
one.  Set  the  instrument  at  D, 
as  near  as  possible  to  C,  and  measure  the  angle  RDL.  It  may  be 
less  than  RCL,  or  greater  than  it,  or  equal  to  it,  according  as  D 
lies  without  the  circle  passing  through  C,  L  and  R,  or  within  it,  or  in 
its  circumference.  The  instrument  should  be  set  as  nearly  as  pos- 
sible in  this  last  position.  To  find  the  proper  correction  for  the 
observed  angle,  observe  also  the  angle  LDC,  (called  the  angle  of 
direction),  counting  it  from  0°  to  360°,  going  from  the  left-hand 
object  toward  the  left ;  and  measure  the  distance  DC.  Calculate 
the  distances  CR  and  CL  with  the  angle  RDL  instead  of  RCL, 
since  they  are  sufficiently  nearly  equal.     Then 


PART  v.] 


TRIAXGILAR  SIRVETIXG. 


269 


RCL  =  RDL  + 


CD  .  sin.  (RDL  +  LDC)        CD  .  sin.  LDC, 


CR  .  sin.  1"  CL  .  sin.l" 

The  last  two  terms  "will  be  the  number  of  seconds  to  be  added 
or  subtracted.  The  Trigonometrical  signs  of  the  sines  must  be 
attended  to.  The  log.  sin.  1"  =4.  6855749.  Instead  of  dividing 
by  sin.  1",  the  correction  without  it,  which  will  be  a  very  small 
fraction,  may  be  reduced  to  seconds  by  multiplying  it  by  2062G5. 
Examine.  Let  RDL  =  32°  20'  18"  .06  ;  LDC  =  101°  15'  32"  .4 ; 
CD  =  0.9;  CR  =  35845.12;  CL  =  29783.1. 

The  first  term  of  the  correction  will  be  +  3". 750,  and  the 
second  term  —  6".113.  Therefore,  the  observed  angle  RDL 
must  be  diminished  by  2". 363,  to  reduce  it  to  the  desired  angle 
RCL. 

Much  calculation  may  be  saved  by  taking  the  station  D  so  that 
all  the  signals  to  be  observed  can  be  seen  from  it.  Then  only  a 
single  distance  and  angle  of  direction  need  be  measured. 

It  may  also  happen  that  the  centre,  C,  of  the         Ffj^  26fi 
signal  cannot  be  seen  from  D.     Thus,  if  the  signal 
be  a  solid  circular  tower,  set  the  Theodohte  at  D, 
and  turn  its  telescope  so  that  its  line  of  sight  be- 
comes tangent  to  the  tower  at  T,  T' ;  measure  on 
these  tangents  equal  distances  DE,  DF,  and  direct 
the  telescope  to  the  middle,  G,  of  the  line  EF.     It 
will  then  point  to  the  centre,  C  ;  and  the  distance  DC  will  equal 
the  distance  from  D  to  the  tower  plus  the  radius  obtained  by  mea- 
suring the  circumference. 

If  the  signa^.  be  rectangular,  measure  DE,  DF. 
Take  any  point  G  on  DE,  and  on  DF  set  oif  DH 

=  DG  ^.     Then  is  GH  parallel   to  EF,    (since 

DG  :  DH  : :  DE  :  DF)    and  the  telescope  directed 
to  its  middle,  K,  will   point  to  the  middle  of  the 

DE 


diagonal  EF.     We  shall  also  Lave  DC  =  DK 


DG' 


Any  such  case  may  be  solved  by  similar  methods. 


For  the  investigation,  see  Appendix  B. 


270  TRIANGULAR  SURVEYLXG.  [part  v 

The  " Phase'^  of  objects  is  the  eflfect  produced  by  the  sub 
ghinino-  on  only  one  side  of  them,  so  that  the  telescope  will  be 
directed  from  a  distant  station  to  the  middle  of  that  bright  side 
instead  of  to  the  true  centre.  It  is  a  source  of  error  to  be  guarded 
against.     Its  effect  may  however  be  calculated. 

(387)  Correction  of  the  angles.  When  all  the  angles  of  any 
triangle  can  be  observed,  their  sum  should  equal  180.*  If  not  they 
must  be  corrected.  If  all  the  observations  are  considered  equally 
accurate,  one-third  of  the  difference  of  their  sum  from  180°,  is  to  be 
added  to,  or  subtracted  from,  each  of  them.  But  if  the  angles  are 
the  means  of  unequal  numbers  of  observations,  their  errors  may  be 
considered  to  be  inversely  as  those  numbers,  and  they  may  be  cor- 
rected by  this  proportion  ;  As  the  sum  of  the  reciprocals  of  each 
of  the  three  numbers  of  observations  Is  to  the  Avhole  error.  So  is 
the  reciprocal  of  the  number  of  observations  of  one  of  the  angles 
To  its  correction.  Thus  if  one  angle  was  the  mean  of  three  obser- 
vations, another  of  four,  and  the  third  of  ten,  and  the  sum  of  all  the 
angles  was  180°  3',  the  first  named  angle  must  be  diminished  by 
the  fourth  term  of  this  proportion  ;  |  +  |  +  ^^  :  3'  : :  |  :  1'  27". 8. 
The  second  angle  must  in  like  manner  be  diminished  by  1'  5". 9  ; 
and  the  third  by  26".8.     Their  corrected  sum  will  then  be  180°. 

It  is  still  more  accurate  but  laborious,  to  apportion  the  total 
error,  or  difference  from  180°,  among  the  angles  inversely  as  the 
"  Weights"  explained  in  Art.  (369) .  On  the  U.  S.  Coast  Survey,  in 
ax  triangles  measured  in  1841  by  Prof.  Bache,  the  greatest'  error 
was  six-tenths  of  a  second. 

(388)  Calculation  and  platting.  The  lengths  of  the  sides  of 
the  triangles  should  be  calculated  with  extreme  accuracy,  in  two 
ways  if  possible,  and  by  at  least  two  persons.  Plane  Trigonometry 
may  be  used  for  even  large  surveys ;  for,  though  these  sides  are 
really  arcs  and  not  straight  lines,  the  difference  will  be  only  ono- 

•  If  the  triangles  were  very  large,  they  would  have  to  be  regarded  as  spherical, 
and  the  sum  of  their  angles  would  be  more  than  180°  ;  but  this  *'  spherical  ex 
ce"ss"  would  be  only  1"  for  a  triangle  containing  76  square  miles,  1  for  4500 
•«5u»re  mil*ii,  &c.;  and  may  therefore  be  neglected  in  all  ordinary  surveying  ope- 
rations. 


PART  v.]  TRUXGlfLlR  SURVEYLXG.  271 

twentieth  of  a  foot  in  a  distance  of  11^  miles  ;  half  a  foot  in  23 
miles  ;  a  foot  in  34^  miles,  &c. 

The  platting  is  most  correctly  done  bj  constructing  the  triangles, 
as  in  Art.  (90),  by  means  of  the  calculated  lengths  of  their  sides. 
If  tlie  measured  angles  are  platted,  the  best  method  is  that  of 
chords,  Art.  (275).  If  many  triangles  are  successively  based  on 
one  another,  they  will  be  platted  most  accurately,  by  referring  all 
their  sides  to  some  one  meridian  line  by  means  of  "  Rectangular  Co- 
ordinates," the  Method  of  Art.  (6),  and  platting  as  in  Art.  (277.) 
In  the  survey  of  a  country,  this  Meridian  would  be  the  true  North 
and  South  hue  passing  through  some  weU  determined  point. 

(389)  Base  of  Verification.  As  mentioned  in  Art.  (380),  a 
side  of  the  last  triangle  is  so  located  that  it  can  be  measured,  as 
was  the  first  base.  If  the  measured  and  calculated  lengths  agree, 
this  proves  the  accuracy  of  all  the  previous  work  of  measurement 
and  calculation,  since  the  whole  is  a  chain  of  which  this  is  the  last 
link,  and  any  error  in  any  previous  part  would  afifect  the  very  last 
line,  except  by  some  improbable  compensation.  How  near  the 
agreement  should  be,  will  depend  on  the  nicety  desired  and  attained 
in  the  previous  operations.  Two  bases  60  miles  distant  differed 
on  one  great  English  survey  28  inches  ;  on  another  one  inch ;  and 
on  a  French  triangulation  extending  over  500  miles,  the  difference 
was  less  than  two  feet.  Results  of  equal  or  greater  accuracy  are 
obtained  on  the  U.  S.  Coast  Survey. 

(390)  Interior  filling  np.  The  stations  whose  positions  have 
been  ietermined  by  the  triangulation  are  so  many  fixed  points, 
froin  which  more  minute  surveys  may  start  and  interpolate  any 
other  points.  The  Trigonometrical  points  are  Hke  the  observed 
Latitudes  and  Longitudes  which  the  mariner  obtains  at  every  oppor- 
tunity, so  as  to  take  a  new  departure  from  them  and  determine 
his  course  in  the  intervals  by  the  less  precise  methods  of  his  com- 
pass and  log.  The  chief  interior  points  may  be  obtained  by  "  Se- 
condary Triangulation,"  and  the  minor  details  be  then  filled  in  by 
any  of  the  methods  of  surveying,  with  Chain,  Compass,  or  Transit; 
already  explained,  or  by  the  Plane  Table,  described  in  Pai-t  VIII. 


272  TRIANGULAR  SURVEYIIVG.  [part  y. 

With  the  Transit,  or  Theodolite,  "  Traversing"  is  the  best  mode  of 
surveying,  the  instrument  being  set  at  zero,  and  being  then 
directed  from  one  of  the  Trigonometrical  points  to  another,  which 
line  therefore  becomes  the  "  Meridian"  of  that  survey.  On  reach- 
ing this  second  point,  in  the  course  of  the  survey,  and  sighting  back 
to  the  first,  the  reading  should  of  course  be  0°,  as  explained  in 
Art.  (377). 

(391)  Radiating  Triangulation.  This  name  may  be  given  t» 
a  method  shown  in  the  figure.  Choose 
a  conspicuous  point,  0,  nearly  in  the 
centre  of  the  field  or  farm  to  be  sur- 
veyed. Find  other  points,  A,  B,  C, 
D,  &c.  such  that  the  signal  at  0  can  be 
seen  from  all  of  them,  and  that  the  tri- 
angles ABO,  BCO,  &c,  shall  be  as 
nearly  equilateral  as  possible.  Mea- 
sure one  side,  AB  for  example.  Ai  A 
measure  the  angles  OAB,  and  OAG ;  at 
B  measure  the  angles  OBA  and  OBC  ;  and  so  on^  around  the 
polygon.  The  correctness  of  these  measurements  may  be  tested 
by  the  sum  of  the  angles,  as  in  Art.  (377).  It  may  also- be  tested 
by  the  Trigonometrical  principle  that  the  product  of  the  sines  of 
every  alternate  angle,  or  the  odd  numbers  in  the  figure,  should 
equal  the  product  of  the  sines  of  the  remaining  angles,  the  even 
numbers  in  the  figure.* 

The  calculations  of  the  unknown  sides  are  readily  made.  In  the 
triangle  ABO,  one  side  and  all  the  angles  are  given  to  find  AO 
and  BO.  In  the  triangle  BCO,  BO  and  all  the  angles  are  given  to 
find  BC  and  CO ;  and  so  with  the  rest.  Another  proof  of  the 
accuracy  of  the  work  will  be  given  by  the  calculation  of  the  length 
of  the  side  AO  in  the  last  triangle,  agreeing  with  its  length  as 
obtained  in  the  first  triangle. 

(392)  Farm  Triangnlation.  A  Farm  or  Field  may  be  surveyed 
by  the  previous  methods,  but  the  following  plan  will  often  be  mw-e 

Foi   the  dt^monstration,  see  Appendix  B. 


PART  v.] 


TRI VXGULAR  SURVEYIXG. 


273 


convenient.  Choose  a  base,  as  XY,  within 

the  field,  and  from  its  ends  measure  the 

angles  between  it  and  the  direction  of 

each  corner  of  the  field,  if  the  Theodo- 

hte  or  Transit  be  used,  or   take   the 

bearing  of  each,  if  the  Compass  be  used. 

Consider  first  the  triangles  which  have 

XY  for  a  base,  and  the  corners  of  the  field.  A,  B,  C,  &c.,  for 

vertices.     In  each  of  them  one  side  and  the  angles  will  be  known  to 

find  the  other  sides,  XA,  XB,  &c.     Then  consider  the  field  as 

made  up  of  triangles  which  have  their  vertices  at  X.     In  each  of 

them  two  sides  and  the  included  angle  will  be  given  to  find  its 

content,  as  in  Art.  (65).     If  Y  be  then  taken  for  the  common 

vertex,  a  test  of  the  former  work  will  be  obtained. 

The  operation  will  be  somewhat  simphfied  by  taking  for  the  base 
.ine  a  diagonal  of  the  field,  or  one  of  its  sides. 


(393)  Inaccessible  Areas.    A  field  or  farm  may  be  surveyed, 
by  this  "  Fourth  Method,"  without  entering  Fig,  270. 

it.  Choose  a  base  line  XY,  from  which  all 
the  corners  of  the  field  can  be  seen.  Take 
their  Bearings,  or  the  angles  between  the 
Base  line  and  their  directions.  The  dis- 
tances from  X  and  Y  to  each  of  them  can 
be  calculated  as  in  the  last  article.  The 
figure  will  then  shew  m  what  manner  the 
content  of  the  field  is  the  difference  between  ^* 
the  contents  of  the  triangles,  having  X  (or  Y)  for  a  vertex,  which 
lie  outside  of  it,  and  those  which  he  partly  within  the  field  and  partl_y 
outside  of  it.  Their  contents  can  be  calculated  as  in  the  last  article, 
and  their  difference  will  be  the  desired  content.  If  the  figure  be 
regarded  as  generated  by  the  revolution  of  a  line  one  end  of  which  is  at 
X,  while  its  other  end  passes  along  the  boundaries  of  the  field,  short- 
ening and  lengthening  accordingly,  and  if  those  triangles  generated 
by  its  movement  in  one  direction  be  called  plus  and  those  generated 
by  the  contrary  movement  be  called  minus,  their  algebraic  sum 
will  be  the  content. 

U8 


274  TRIUVGULAR  SURFEYIXG.  [paet  v 

(394)  Inversion  of  the  Fourth  Method.  In  all  the  opera- 
tions  which  have  been  explained,  the  position  of  a  point  has  been 
determined,  as  in  Art.  (8),  by  taking  the  angles,  or  bearings,  of 
two  lines  passing  from  the  two  ends  of  a  Base  line  to  the  unknown 
point.  But  the  same  determination  may  be  effected  inversely,  by 
taking  from  the  point  the  bearings,  by  compass,  of  the  two  ends  of 
the  Base  Hne,  or  of  any  two  known  points.  The  unknown  point 
will  then  be  fixed  by  platting  from  the  two  known  points  the  oppo- 
site bearings,  for  it  will  be  at  the  intersection  of  the  lines  thus 
determined. 

(395)  Defects  of  the  Method  of  Intersection.  The  determi- 
nation of  a  point  by  the  Fourth  Method  (enunciated  in  Art.  (8), 
and  developed  in  this  Part)  founded  on  the  intersection  of  lines, 
has  the  serious  defect  that  the  point  sighted  to  will  be  very  indefi- 
nitely determined  if  the  fines  which  fix  it  meet  at  a  very  acute  or 
a  very  obtuse  angle,  which  the  relative  positions  of  the  points  observed 
from  and  to,  often  render  unavoidable.  Intersections  at  right 
angles  should  therefore  be  sought  for,  so  far  as  other  considerations 
will  permit. 


PAUT  VT. 


TRILINEAR   SURVEYING; 

By  the  Fifth  Method. 

(396)  Trilinear  Surveying  is  founded  on  the  Fifth  Method  of 
determining  the  position  of  a  point,  by  measuring  the  angles  betwen 
three  lines  conceived  to  pass  from  the  required  point  to  three 
known  points,  as  illustrated  in  Art.  (10). 

To  fix  the  place  of  the  point  from  these  data  is  much  more  diffi- 
cult than  in  the  preceding  methods,  and  is  known  as  the  "  Problem  of 
the  three  points."  It  will  be  here  solved  Geometrically,  Instni- 
mentally  and  Analytically. 


(397)  Geometrical  Solution.    Let  A,  B  and  C  be  the  known 

Fis.  271. 


objects  observed  from  S,  the  angles  ASB  and  BSC  being  there 
measured.  To  fix  this  point,  S,  on  the  plat  containing  A,  B  and 
C,  draw  lines  from  A  and  B,  making  angles  with  AB  each  equal 


276  TRILIXEAR  SURVEYIAG.  [pari  vi 

to  90° — ASB.  The  intersection  of  these  Imes  at  0  will  be  the 
centre  of  a  circle  passing  through  A  and  B,  in  the  circumference 
"of  which  the  point  S  will  be  situated.*  Describe  this  circle.  Also, 
draw  lines  from  B  and  C,  making  angles  with  BC,  each  equal  to 
90=  —  BSC.  Their  intersection,  0',  will  be  the  centre  of  a  circle 
passing  through  B  and  C.  The  point  S  wiU  lie  somewhere  in  its 
circumference,  and  therefore  in  its  mtersection  with  the  fornjer 
circumference.     The  point  is  thus  determined. 

In  the  figure  the  observed  angles,  ASB  and  BSC,  are  supposed 
to  have  been  respectivelj'  40°  and  60°.  The  angles  set  off  are 
therefore  50°  and  30°.  The  central  angles  are  consequently  80° 
and  120°,  twice  the  observed  angles. 

The  dotted  lines  refer  to  the  checks  explained  in  the  latter  part 
of  this  article. 

When  one  of  the  angles  is  obtuse,  set  off  its  difference  from  90° 
on  the  opposite  side  of  the  line  joining  the  two  objects  to  that  on 
which  the  point  of  observation  lies. 

When  the  angle  ABC  is  equal  to  the  supplement  of  the  sum  of 
the  observed  angles,  the  position  of  the  point  will  be  indeterminate  ; 
for  the  two  centres  obtained  will  coincide,  and  the  circle  described 
from  this  common  centre  will  pass  through  the  three  points,  and 
any  point  of  the  circumference  will  fulfil  the  conditions  of  the  prob- 
lem. 

A  third  angle,  between  one  of  the  three  points  and  a  fourth 
point,  should  always  be  observed  if  possible,  and  used  like  the 
others,  to  serve  as  a  check. 

Many  tests  of  the  correctness  of  the  position  of  the  point  deter- 
mined may  be  employed.  The  simplest  one  is  that  the  centres  of 
the  circles,  0  and  0',  should  lie  in  the  perpendiculars  drawn  through 
the  middle  points  of  the  lines  AB  and  BC. 

Another  is  that  the  line  BS  should  be  bisected  perpendicularly 
by  the  line  00'. 

A  third  check  is  obtained  by  drawing  at  A  and  C  perpendiculars 
to  AB  and  CB,  and  producing  them  to  meet  BO  and  BO'  produced, 

For,  tbe  arc  AB  measures  the  angle  AOB  at  the  centre,  which  angle  =  180' 
•-2(900  — ASB)  =2  ASB.  Therefore,  any  angle  inscribed  in  the  ciicumfer 
ence  and  measured  by  the  same  arc  is  equal  to  ASB 


PART  VI.]  TRILINEAR  SURVEYING.  277 

in  D  and  E.  The  line  DE  should  pass  through  S ;  for,  the  angles 
BSD  and  BSE  being  right  angles,  the  lines  DS  and  SE  form  one 
straight  line. 

The  figure  shews  these  three  checks  by  its  dotted  lines. 

(398)  Instrumental  Solution.  The  preceding  process  is  tedious 
where  many  stations  are  to  be  determined.  They  can  be  more 
readily  found  by  an  instrument  called  a  Station-pointer,  or  Choro- 
grapli.  It  consists  of  three  arms,  or  straight-edges,  turning  about 
a  common  centre,  and  capable  of  being  set  so  as  to  make  with  each 
other  any  angles  desired.  This  is  effected  by  means  of  graduated 
arcs  carried  on  their  ends,  or  by  taking  off  with  their  points  (as 
with  a  pair  of  dividers)  the  proper  distance  from  a  scale  of  chords 
(see  Art.  (274))  constructed  to  a  radius  of  their  length.  Being 
thus  set  so  as  to  make  the  two  observed  angles,  the  instrument  is 
laid  on  a  map  containing  the  three  given  points,  and  is  turned 
about  till  the  three  edges  pass  through  these  points.  Then 
their  centre  is  at  the  place  of  the  station,  for  the  three  points  there 
subtend  on  the  paper  the  angles  observed  in  the  field. 

A  simple  and  useful  substitute  is  a  piece  of  transparent  paper, 
cr  ground  glass,  on  which  three  lines  may  be  drawn  at  the  proper 
ingles  and  moved  about  on  the  paper  as  before. 

(399)  Analytical  Solution.  The  distances  of  the  required 
point  from  each  of  the  known  points  may  be  obtamed  analytically. 
Let  AB  =  i?;  BC  =  a;  ABC  =  B;  ASB  =  S;  BSC  =  S'.  Also, 
make  T  =  360°  —  S  —  S'  —  B.  Let  BAS  =  U ;  BCS  =  V. 
Then  we  shall  have  (as  will  be  shewn  in  Appendix  B) 

Cot.  U  =  cot.  T  {-±1^1^—-  +  l) 
\a  .  sm.  S  .  COS.  T         / 

V  =  T  — U 

c-D       ^  •  sin.  U  a  .  sin.  V 

b±$  =  — : — -— :  or,  =  — : — -- — 

sm.  S     '      '  sm.  S' 

g  .  _  r  .  sin.  ABS         j^p  _a  .  sin.  CBS 
sin.  S      *  ~~      sin.  S' 


278  TRILINEIR  SURVEYING.  [part  vi 

Attention  must  be  given  to  the  algebraic  signs  of  the  trigonome- 
trical functions. 

Example.  ASB  =  33°  45' ;  BSC=  22°  30';  AB  =  600  feet ; 
BC  =  400  feet ;  AC  =  800  feet.  Required  the  distances  and 
directions  of  the  point  S  from  each  of  the  stations. 

In  the  triangle  ABC,  the  three  sides  being  known,  the  angle 
ABC  is  found  to  be  104"  28'  39".  The  formula  then  gives  the 
angle  BAS  =  U  =  105°  8'  10"  ;  whence  BCS  is  found  to  be  94° 
8'  11" ;  and  SB  =  1042.51 ;  SA  =  710.198  ;  and  SC  =  934.291. 

(400)  Maritime  Surveying.  The  chief  application  of  the  Tri- 
Unear  Method  is  to  Maritime  or  Hydro  graphical  Surveying,  the 
object  of  which  is  to  fix  the  positions  of  the  deep  and  shallow  points 
in  harbors,  rivers,  &c.,  and  thus  to  discover  and  record  the  shoals, 
rocks,  channels  and  other  important  features  of  the  locality.  To 
effect  this,  a  series  of  signals  are  established  on  the  neighboring 
shore,  any  three  of  which  may  be  represented  by  our  points  A,B,  C. 
They  are  observed  to  from  a  boat,  by  means  of  a  sextant,  and  the 
position  of  the  boat  is  thus  fixed  as  just  shewn.  The  boat  is  then 
rowed  in  any  desired  direction,  and  soundings  are  taken  at  regular 
intervals,  till  it  is  found  convenient  to  fix  the  new  position  of  the 
boat  as  before.  The  precise  point  where  each  sounding  was  taken 
can  now  be  platted  on  the  map  or  chart.  A  repetition  of  this  pro- 
cess will  determine  the  depths  and  the  places  of  each  point  of  the 
bottom. 


PAET  VII 

OBSTACLES  IN  ANGULAR  SURVEYING. 

(401)  The  obstacles,  such  a*  trees,  houses,  hills,  vallies,  rivera, 
&c.,  which  prevent  the  direct  alinement  or  measurement  of  any 
desired  course,  can  be  overcome  much  more  easily  and  precisely 
with  any  angular  instrument  than  with  the  chain,  methods  for  using 
wliich  were  explained  in  Part  II,  Chapter  V.  They  will  however 
be  taken  up  in  the  same  order.*  As  before,  the  given  and  measured 
lines  are  drawn  with  fine  full  lines ;  the  visual  lines  with  broken 
lines ;  and  the  lines  of  the  result  with  heavy  full  lines. 

CHAPTER  I. 

PERPENDICULARS  AlVD  PARALLELS. 

(102)  Erecting  Perpendiculars,  To  erect  a  'perpendicular  to 
a  line  at  a  given  point,  set  the  instrument  at  the  given  point,  and, 
if  it  be  a  Compass,  direct  its  sights  on  the  line,  and  then  turn  them 
till  the  new  Bearing  differs  90°  from  the  original  one,  as  explained 
m  Art.  (243).  A  convenient  approximation  is  to  file  notches  in 
the  Compass-plate,  at  the  90°  points,  and  stretch  over  them  a  thread, 
sighting  across  which  will  give  a  perpendicular  to  the  du-ection  of 
the  sights. 

The  Transit  or  Theodolite  being  set  as  above,  note  the  readmg 
of  the  vernier  and  then  turn  it  till  the  new  reading  is  90°  more  oi 
less  than  the  former  one. 

The  Demonsti-ations  of  the  Problems  which  require  thpm,  and  from  which 
they  can  conveniently  be  separated,  will  be  found  in  App         v  B. 


280 


OBSTACLES  IN  ANGULAR  SURVEYIIVG.     [part  vn 


(403)  To  erect  a  perpendicular  to  an  inaccessible  line,  at  a 
given  point  of  it.     Let  AB  be  the  line  Fig.  272. 

and  A  the  point.  Calculate  the  distance 
from  A  to  any  point  C,  and  the  angle 
CAB,  by  the  method  of  Art.  (430) .  Set 
the  instrument  at  C,  sight  to  A,  turn  an 
angle  =  CAB,  and  measure  in  the  direc- 
tion thus  obtained  a  distance  CP  =  CA  .  cos.  CAB. 

the  required  perpendicular,  y// 
n  }  - 


PA  wiU  be 


(404)  Letting  fall  perpendiculars.  To  let  fall  a  perpendi- 
cular to  a  line  from  a  given  point.  With  the  Comjjass,  take  the 
Bearing  of  the  given  line  and  then  from  the  given  point  run  a  line, 
with  a  Bearing  differing  90°  from  the  original  Bearing,  till  it  reaches 
the  given  line. 

With  the  Transit  or  Theodolite,  set  it  at  any  point  of  the  given 
line,  as  A,  and  observe  the  angle  between  this 

line  and  a  line  thence  to   the  given    point,    A\; 

P.     Then  set  at  P,  sight  to  the  former  posi-  \ 

tion  of  the  instrument,  and  turn  a  number  of 

degrees  equal  to  what  the  observed  angle  at 

A  w^anted  of  90°.     The  instrument  will  then 

point  in  the  direction  of  the  required  perpendicular  PB. 


Fig.  273. 


X 


(405)  To  let  fall  a  perpendicular  to  a  line  from  aninaccesdibU 
■point.  Let  AB  be  the  hne  and  P  the 
point.  Measure  the  angles  PAB,  and 
PBA.  Measure  AB.  The  angles  APC 
and  BPC  are  known,  bemg  the  comple- 
ments of  the  angles  measured.  Then  is 
tan.  APC 


AC  =  AB 


tan.  APC  +  tan.  BPC" 


ht^L 


f^r  f>:- 


t^bCHt^'j^t. 


CHAP,  i]  Perpendiculars  and  Parallels.  28J 

(406)  To  let  fall  a  perpendicular  to  an  inaccessible  line  from 
a  giveyi  point.     Let  C  be  the  point  and  Fig.  275. 

AB  the  line.  Calculate  the  angle  CAB 
by  the  method  of  Art.  (430).  Set  the 
instrument  at  C,  sight  to  A,  and  turn  an 
angle  =  90  —  CAB.  It  will  then  pomt 
in  the  direction  of  the  required  perpen- 
dicular CE. 

(407)  Running  Parallels.  To  trace  a  line  through  a  given 
point  parallel  to  a  given  line.  With  the  Compass,  take  the  Bear- 
ing of  the  given  line,  and  then,  from  the  given  point,  run  a  line 
with  the  same  Bearing. 

With  the  Transit  or  Theodolite,  set  it  at  any  convenient  x)oint 
of  the  given  Hne,  aa  A,  direct  Fig-  276. 

it  on  this  line,  and  note  the  read-  ■^\ ^ 

ing.     Then  turn  the  vernier  till        \ 
the  cross-hairs  bisect  the   given 

point,  P.     Take  the  instrument  to  p  Q 

this  point  and  sight  back  to  the  former  station,  by  the  lower  motion, 
without  changing  the  reading.  Then  move  the  vernier  till  the 
reading  is  either  the  same  as  it  was  when  the  telescope  was  di- 
rected on  the  given  Une,  or  is  180°  different.  It  will  then  be  di- 
rected (forward  or  backward)  on  PQ,  a  parallel  to  AB,  since  equal 
angles  have  been  measured  at  A  and  P.  The  manner  of  reading 
them  is  similar  to  the  method  of  "Traversing,"  Art.  (373). 

(408)  To  trace  a  line  through  a  given  point  parallel  to  an 
iiiaccessible  line.     Let  C  be  the  given  Fig.  277. 
point,  and  AB  the   inaccessible   line. 
Find  the  angle  CAB,  as  in  Art.  (430). 
Set  the  instrument  at  C,  direct  it  to  A, 

and  then  turn  it  so  as  to  make  an  angle  C  E 

with  CA  equal  to  the  supplement  of  the  angle  CAB.  It  will  then 
poiat  in  a  direction,  CE,  parallel  to  AB. 


282  OBSTACLES  IN  lAGlXAR  SURVEYING.      [pari  m 


CHAPTER  II. 


OBSTACLES  TO  ALINEIIENT. 


A.   To   PROLONG  A   LINE. 


(409)  The  mstrument  being  set  at  the  farther  end  of  a  line,  and 
directed  back  to  its  beginning,  the  sights  of  the  Compass^  if  that 
be  used,  will  at  once  give  the  forward  direction  of  the  line.  They 
serve  the  purpose  of  the  rods  described  in  Art.  (169).  A  distant 
pouit  being  thus  obtamed,  the  Compass  is  taken  to  it  and  the  pro- 
cess repeated.  The  use  of  the  Transit  or  Theodolite,  for  this 
purpose,  was  fully  explained  in  Art.  (376). 

(410)  By  perpendiculars.  AVhen  a  tree,  or  house,  obstructing 
the  line,  i?  met  with,  place  the  instru-  Fig-  278. 

ment  at  a  point  B  of  the  Hue,  and  set  ^ 
off  there  a  perpendicular,  to  C  ;  set  off 
another  at  C  to  D,  a  third  at  D  to  E, 
making  DE  =  BC,  and  a  fourth  at  E,  which  last  will  be  in  the 
direction  of  AB  prolonged.  If  perpendiculars  cannot  be  con- 
veniently used,  let  BC  and  DE  make  any  equal  angles  with 
the  line  AB,  so  as  to  make  CD  parallel  to  it. 

(411)  By  an  equilateral  triangle. 

At  B,  turn  aside  from  the  Hne  at  an 
angle  of  60°,  and  measure  some  con- 
venient distance  BC.  At  C,  turn  60" 
in  the  contrary  direction,  and  mea-  c 

sure  a  distance  CD  =  BC.  Then  will  D  be  a  point  in  the  line  AB 
prolonged.  At  D,  turn  60°  from  CD  prolonged,  and  the  new 
direction  will  be  in  the  line  of  AB  prolonged.  This  method  re 
quires  the  measurement  of  one  angle  less  than  the  preceding. 


euAP.  II.] 


Obstacles  to  ilinement. 


283 


(412)  By  triangulation.  Let  Fig  28o. 
AB  be  the  line  to  be  prolonged. 
Choose  some  station  C,  whence 
can  be  seen  A,  B,  and  a  point 
beyond  the  obstacle.  Measure 
AB  and  the  angles  A  and  B,  of 
the  triangle  ABC,  and  thence  calculate  the  side  AC.  Set  the 
instrument  at  C,  and  measure  the  angle  ACD,  CD  being  any  Hne 
which  will  clear  the  obstacle.  Let  E  be  the  desired  point  in  the 
lines  AB  and  CD  prolonged.  Then  in  the  triangle  ACE,  will  be 
known  the  side  AC  and  its  including  angles,  whence  CE  can  be 
calculated.  Measure  the  resulting  distance  on  the  ground,  and 
its  extremity  will  be  the  desired  point  E.  Set  the  instrument  at 
E,  sight  to  C,  and  turn  an  angle  equal  to  the  supplement  of  the 
angle  AEC,  and  you  will  have  the  direction,  EF,  of  AB  prolonged 

(413)  When  the  line  to  be  prolonged  is  inaccessible.      In 

this  case,  before  the  preceding  method  can  be  apphed,  it  will  be 
necessary  to  determine  the  lengths  of  the  lines  AB  and  AC,  and 
the  angle  A,  by  the  method  given  in  Art.  (430). 

(414)  To  prolong  a  line  with  only  an  angular  instrnment. 

This  may  be  done  when  no  means  of  measuring  any  distance  can  be 
obtained.     Let  AB  be  the  Une  Fig.  281. 
to  be  prolonged.     Set  the  in-  C 
strument  at  B  and  deflect  an- 
gles of  45°  in  the  directions  C  4_ B, 

and  D.  Set  at  some  point,  C, 
©n  one  of  these  lines  and  deflect 
from  CB  45°,  and  mark  the 
point  D  where  this  direction  intersects  the  direction  BD.  Also,  at 
0,  deflect  90°  from  CB.  Then,  at  D,  deflect  90°  from  DB.  The 
mtersections  of  these  last  directions  will  fix  a  point  E.  At  E 
deflect  135°  from  EC  or  ED,  and  a  line  EF,  m  the  direction  of 
AB  will  be  obtained  and  may  be  continued.* 


*  This  ingenious  conti-ivance  is  due  to  a  former  student,  Mr.  R.  Hood,  in  wlioso 
practice,  while  running  an  air  line  for  a  raih'oad,  the  necessity  occurred. 


284  OBSTACLES  IN  ANGULAR  SURVEYING,    [part  vh 

B.    To   INTERPOLATE   POINTS    IN   A   LINE. 

(415)  The  instrument  being  set  at  one  end  of  a  line  and  du-ected 
to  the  other,  intermediate  points  can  be  found  as  in  Art.  (177), 
&c.  If  a  valley  intervenes,  the  sights  of  the  Compass,  (if  the 
Compass-plate  be  very  carefully  kept  level  cross-ways),  or  the  tele- 
scope of  the  Transit  or  Theodolite,  answer  as  substitutes  for  the 
plmnb-hue  of  Art.  (179). 

(416)  By  a  random  line.  When  a  wood,  hill,  or  other  obstar 
cle,  prevents  one  end  of  the  line,  Z,  Fig.  282. 

from  bemg  seen  from  the  other.  A,  run 
a  random  line  AB  with  the  Compass  or 
Transit,  &c.,  as  nearly  in  the  desired 
direction  as  can  be  guessed,  till  you  arrive  opposite  the  point  Z. 
Measure  the  error,  BZ,  at  right  angles  to  AB,  as  an  offset.  Multi- 
ply this  error  by  57 j%,  and  divide  the  product  by  the  distance  AB. 
The  quotient  will  be  the  degrees  and  decimal  parts  of  a  degree, 
contained  in  the  angle  BAZ.  Add  or  subtract  this  angle  to  or 
from  the  Bearing  or  reading  with  which  AB  was  run,  according  to 
the  side  on  which  the  error  was,  and  start  from  A,  with  this  cor- 
rected Bearing  or  reading,  to  run  another  line,  which  will  come  out 
at  Z,  if  no  error  has  been  committed.* 

Example.  A  random  hne  was  run,  by  compass,  with  a  Bearing 
of  S.  80°  E.  At  20  chains'  distance  a  point  was  reached  opposite 
to  the  desired  point,  and  10  links  distant  from  it  on  its  right. 
Required  the  correct  Bearing. 

Ans.     By  the  rule,  ^^  ^  ^^°^  =  0°.2865  =  17'.      The  cor- 
^  '        2000 

rect  Bearing  is  therefore  S.  80°  17'  E.     If  the  Transit  had  been 

used,  its  reading  would  have  been  changed  for  the  new  line  by  the 

same  17'.    A  simple  diagram  of  the  case  will  at  once  shew  whether 

the  correction  is  to  be  added  to  the  original  Bearing  or  angle,  or 

subtracted  from  it. 

*  This  rule  ia  substantially  identical  with  that  of  Art.  (319),  whore  its  reason  ia 
given 


CHAP.  11  ~  Obstacles  to  ilincment.  285 

K  Trigonometrical  Tables  are  at  hand,  the  correction  will  be 
more  precisely  obtained  from  tliis  equation ;    Tan.  BAZ  ^  — . 

In  this  example,   — ^  =  — -—  =  .005  =  tan.  17'. 

The  57°. 3  rule,  as  it  is  sometimes  called,  may  be  variously  modi- 
fied. Thus,  multiply  the  error  by  86°,  and  divide  by  one  and  a  half 
times  the  distance  ;  or,  to  get  the  correction  in  minutes,  multiply 
by  3438  and  divide  by  the  distance  ;  or,  if  the  error  is  given  in 
feet  and  the  distance  in  four-rod  chains,  multiply  the  former  by  52 
and  divide  by  the  distance,  to  get  the  correction  in  minutes. 

The  correct  line  may  be  run  -with  the  Bearing  of  the  random 
line,  by  turning  the  vernier  for  the  correction,  as  in  Art.  (312). 

(417)  By  Latitudes  and  Departures.      When        Fig.  283. 
a  single  line,  such  as  AB,  cannot  be  run  so  as  to      A 

come  opposite  to  the  given  point  Z,  proceed  thus,    ^1 z 

with  the  Compass.     Run  any  number  of  zig-zag        !     ^^'Ai 

courses,    AB,    BC,  CD,  DZ,  in    any  convenient       I  ^^"^^^'d 

direction,  so  as  at  last  to  arrive  at  the  desired  point.       I   0^y\ 

Calculate  the  Latitude  and  Departure  of  each  of 

these  courses  and  take  their  algebraic  sums.     The 

sum  of  the  Latitudes  will  be  equal  to  AX,  and  that 

XZ 
of  the  Departures  to  XZ.  Then  is  Tan.  ZAX  =  ^=—  ; 

XA 

1.  e.  the  algebraic  sum  of  the  Departures  divided 

by  the  algebraic  sum  of  the  Latitudes  is  equal  to  the  tangent  of 

the  Bearing.* 

(418)  When  the  Transit  or  Theodolite  is  used,  any  hne  may  be 
taken  as  a  Meridian,  i.  e.  as  the  line  to  which  the  followino-  Unea 
are  refened;  as  in  "Traversing,"  Art.  (373),  page  254,  all  the 
successive  lines  were  referred  to  the  first  line.  In  the  figure,  on 
the  next  page,  the  same  lines  as  in  the  preceding  figure  are  repr©* 

The  length  of  the  line  AZ  can  also  oe  at  once  obtained  since  it  is  equal  to 
the  square  root  of  the  snm^of  the  squares  of  AX  and  XZ  ;  or  to  the  Latituda 
divided  by  the  cosine  of  the  Bearing. 


286 


OBSTACLES  L\  AXGPLAR  SFRVEYIIVG,    [part  vn. 


sented,  but  they  are  referred  to  the  first  course, 
AB,  instead  of  to  the  Magnetic  Meridian  as 
before,  and  their  Latitudes  are  measured  along 
its  produced  line,  and  its  Departures  perpen- 
dicular to  it.  As  before,  a  right-angled  triangle 
will  be  formed,  and  the  angle  ZAY  will  be 
the  angle  at  A  between  the  first  line  AB  and 
the  desired  line  AZ. 

This  method  of  operation  has  many  useful^ 
applications,  such  as  in  abtaining  data  for  running  Railroad  CurveSj 
&c.,  and  the  student  should  master  it  thoroughly. 

The  desired  angle  (and  at  the  same  time  the  distance)  can  be 
obtained,  approximately,  in  this  and  the  preceding  case,  by  finding 
in  a  Traverse  Table,  the  final  Latitude  and  Departure  of  the  desired 
line  (or  a  Latitude  and  Departure  having  the  same  ratio)  and  the 
Bearing  and  Distance  corresponding  to  these  will  be  the  angle 
and  distance  desu'ed. 


(419)  By  similar  triangles. 

Through  A  measure  any  line  CD. 
Take  a  point  E,  on  the  line  CB, 
beyond  the  obstacle,  and  from  it 
set  oflF  a  parallel  to  CD,  to  some 
point,  F,  in  the  line  DB.  Measure 
EF,  CD,  and  CA.  Then  this  pro- 
portion, CD  :  CA  : :  EF 
E  to  a  point  in  the  line  AB. 


:^ 


EG,  will  give  the  distance  EG,  from 
So  for  other  points. 


Fi-.  2£ 


(420)  By  triangnlation.  When 
obstacles  prevent  the  preceding  me- 
thods being  used,  if  a  point,  C,  can  be 
found,  from  which  A  and  B  are  accessi- 
ble, measure  the  distances  CA,  CB, 
and  the  angle  ACB,  and  thence  calculate  the  angle  CAB.  Then 
observe  any  angle  ACD,  beyond  the  obstacle.  In  the  triangle 
ACD,  a  side  and  its  including  angles  are  known,  to  find  CD.  Mea* 
Bure  it,  and  a  point,  D,  in  the  desired  line,  will  be  obtained. 


CHAP.  III.]  Obstacles  to  Measurement.  287 


'■^ 


\f^  \  CHAPTER  III. 


OBSTACLES  TO  MEASUREMENT. 

A.  When  both  ends  of  the  line  are  accessible. 

(421)  The  methods  given  in  the  preceding  Chapter  for  prolong- 
hig  a  line  and  for  interpolating  points  in  it,  will  generally  give  tha 
length  of  the  line  by  the  same  operation.  Thus,  in  Fig.  278,  the 
maccessible  distance  BE  is  equal  to  CD  ;  in  Fig.  279,  BD  =  BC 
=  CD  ;  in  Fig.  280,  the  distance  BE  can  be  calculated  from  the 
same  data  as  CE ;  m  Fig.  282,  AZ  =  V(AB3  +  BZ^)  ;  in  Fig. 
283,  AZ  =  V(AX=^  +  XZ2)  ;  in  Fig.  284,  AZ  =  V(AY2  + 

YZ2)  ;  in  Fig.  285,  AG  =  —  ^^f"  ~  ^^^ ;  in  Fig.  286,  the  tri- 

angle  ACD  will  give  the  distance  AD.  The  method  of  Latitudes 
and  Departures,  Arts.  (417)  and  (418),  is  very  generally  appli- 
*^ftble.     So  is  the  following.  * 

(422)  By  triangulation.  Let  AB  Fig.  287. 
be  the  inaccessible  distance.  From 
any  point,  C,  from  which  both  A  and 
B  are  accessible,  measure  CA,  CB, 
and  the  angle  ACB.  Then  in  the 
triangle  ABC  two  sides  and  the  in- 
cluded angle  are  known  to  find  the 
Bide  AB.  If  all  the  angles  can  be  measured,  they  may  be  coP' 
rected,  as  in  Art.  (387)-* 

(423)  A  l)roken  Base.  When  the  angle  C  is  very  obtuse,  the 
preceding  problem  may  be  modified  as  follows.  Naming  the  lines 
AS  is  usual  in  Trigonometry,  by  small  letters  corresponding  to  the 

•  In  this  figm-e,  and  tlie  following  ones,  the  angular  point  ennloaed  ia  a  circl« 
indicates  the  place  at  which  the  instrument  is  set 


288  OBSTACLES  Ii\  ANGULAR  SURVEYDG.     [part  vn 

capital  letters  at  the  angles  to  which  they  are  opposite,  and  letting 
K  =  the  number  of  minutes  in  the  supplement  of  the  angle  C,  we 

Fig.  288. 


^7 

c 

shall  have 

abK^ 


AB  =  c  =  a  +  b  —  0.000000042308  x 


a+b' 

This  formula  is  chiefly  used  in  the  case  of  what  is  called  in  Tri- 
angular Surveying  "  A  broken  Base  ;"  such  as  above ;  AC  and 
CB  being  measured  and  forming  very  nearly  a  straight  hue,  and 
the  length  of  AB  being  required. 

Log.  0.000000042308  =  2.6264222  —  10. 

(424)  By  angles  to  known  points.  The  length  of  a  line,  both 
ends  of  which  are  accessible,  may  also  be  determined  by  angles 
measured  at  its  extremities  between  it  and  the  directions  of  two  or 
more  known  points.  But  as  the  methods  of  calculation  involve  sub- 
sequent problems,  they  will  be  postponed  to  Articles  (435),  (436) 
and  (437). 

B.   When  one  end  op  the  line  is  inaccessible. 

(425)  By  perpendiculars.  Many  of  the  methods  given  for 
the  chain,  in  Part  II,  Chapter  Y,  may  be  still  more  advantageously 
employed  with  angular  instruments,  which  can  so  much  more  easily 
and  precisely  set  oflf  the  Perpendiculars  required  in  Articles  (191), 
(192),  (193),  &c. 

(426)  By  equal  angles.    LetAB  Fig.  289. 

be  the  inaccessible  line.    At  A  set  off  3)^ 

AC,  perpendicular  to  AB,  and  as       ""^s 

nearly  equal  to  it,  by  estimation,  as  ^"^n 

the  ground  will  permit.     At  C,  mea-  ^^  ..^, 

sure  the  angle  ACB,  and  turn  the  C  |i 

sights,  or  vernier,  till  ACD  =  ACB. 

Find  the  point,  D,  at  the  intersections  of  the  lines  CD  and  BA  pro* 

^vid.     Then  is  AD  =  AB. 


CHAP.   III.] 


Obstacles  to  Measarement< 


289 


(427)  By  triaogulation.     Measure  a  distance         Fig.  29o. 

AC,  about  equal  to  AB.     Measure  the  angles  at 

A  and  C.    Then  in  the  triangle  ABC,  two  angles 

and  the  included  side  are  known,  to  find  another 

. ,      .  -D       AC  sin.  ACB 

side,  AB  = — . 

'  sin.  ABC 

When  the  compass  is  used,  the  angles  between 
die  lines  will  be  deduced  from  their  respective 
Bearings,  by  the  principles  of  Art.  (243). 

If  the  angle  at  A  is  90%  AB  =  AC  .  tang.  ACB. 

If  A  =  90°,  and  C  =  45°,  then  AC  =  AB ;  but  this  position 
could  not  easily  be  obtained,  except  by  the  use  of  the  Sextant, 
a  reflecting  instrument,  not  described  in  tliis  volume. 


(428)   When  one  point  cannot  be  seen  from  the  other.— 

Choose  two  points,  C  and  D,  in  the  line 
of  A,  and  such  that  from  C,  A  and  B  can 
be  seen,  and  from  D,  A  and  B.  Measure 
AC,  AD,  and  the  angles  C  and  D.  Then, 
in  the  triangle  BCD,  are  known  two  an- 
gles and  the  included  side,  to  find  CB. 
Then,  in  the  triangle  ABC,  are  known 
two  sides  and  the  included  angle,  to  find 
the  third  side,  AB. 


(429)  To  find  the  distance  from  a  given  point  to  an  inacces- 
sible line.  In  Fig.  275,  Art.  (406),  the  required  distance  is 
CE.  The  operations  therein  directed  give  the  line  CA  and  the 
angle  CAB  or  CAE.    The  required  distance  CE  =  CA  .  sin.  CAE. 


19 


290 


OBSTACLES  IN  ANGULAR  SURVEYING.      [pakt  vn 


C.  When  both  ends  op  the  line  are  inaccessible. 

(430)  General  Method.    Let  f^s-  292. 

A.B  be  the  inaccessible  line. 
Measure  any  convenient  distance 
CD,  and  the  angles  ACD,  BCD, 
ADC,  BDC. 

Then,  in  the  triangle  CDA, 
two  angles  and  the  included  side 
are  given,  to  find  CA.  In  the 
triangle  CDB,  two  angles  and  the 
included  side  are  given,  to  find 
CB.  Then,  in  the  triangle  ABC, 
two  sjdes  and  the  included  angle 
are  given,  to  find  AB. 

The  work  may  be  verified  by  taking  another  set  of  triangles, 
and  finding  AB  from  the  triangle  ABD  instead  of  ABC. 

The  following  formulas  will  however  give  the  desired  distance 

with  less  labor. 

™    ,  .    T-         1,1.,  TT       sin.  ADC  .  sin.  CBD 

Fmd  an  angle  K,  such  that  tang.  K  =  - — j:^—— — : — stttt' 
°  °  sm,  CAD  .  sm.  BDC 

Then  find  the  difference  of  the  unknown  angles  in  the  triangle 
CAB  from  the  formula 

Tang.  1  (CAB  — ABC)  =  tang.  (45°  — K)  .  cot.  |  ACB. 
Then  is  CAB  =  |  (CAB  —ABC)  +  I  (CAB  +  ABC). 

,,.  „   .  -r,   r^T\  sin.  BDC  .  sin.  ACB 
Fmally,  AB  =  CD  - — j^^=- — : — prr^. 
•^ '         sm.  CBD  .  sm.  CAB 

Example.  Let  CD  =  7106.25  feet;  ACD  =  95°  17'  20"; 
BCD  =  61°  41'  50" ;  ADC  =  39''  38'  40" ;  BDC  =  78°  35'  10"; 
required  AB. 

The  figure  is  constructed  with  these  data  on  a  scale  of  5000  feet 
to  1  inch  =  1:60000. 

By  the  above  formulas,  K  is  found  to  be  30'  26'  5'  ;  CAB  = 
113°  55'  37"  ;  and  lastly  AB  =  6598.32. 

Both  the  methods  may  be  used  as  mutual  checks  in  any  im 
portant  case. 


CHAP.  III."! 


Obstacles  to  Measurement. 


291 


If  the  lines  AB  and  CD  crossed 
each  other,  as  in  Fig.  293,  instead  of 
being  situated  as  in  the  preceding 
figure,  the  same  method  of  calcular 
tion  would  apply. 


(431)  Probiem.  To  measure  an  inaccessible  distance,  AB, 
when  a  'point,  C,  in  its  line  can  be  obtained.  Set  the  instrument 
at  a  point,  D,  from  which  A,  B 
and  C  can  be  seen,  and  measure 
the  andes  CDA  and  ADB. 
Measure  also  the  Une  DC  and 
the  angle  C.  Then  in  the  tri- 
angle ACD  two  angles  and  the 
included  side  are  given  to  find 
AD.  In  the  triangle  DAB,  the 
angle  DAB  is  known,  (being  equal  to  ACD  4-  CDA),  and  AD 
having  been  found,  we  again  have  two  angles  and  the  included  side 
to  find  AB. 


(432)  Problem.  To  measure  an  inaccessible  distance,  AB, 
when  only  one  point,  C,  can  be  found  from  which  both  ends  of  the 
line  can  be   seen.     Consider  CA  Fi".  29.5. 

and  CB  as  distances  to- be  deter-        ^rr -2^ 

mined,  having  one  end  accessible. 
Determine  them,  as  in  Art.  (427), 
by  choosing  a  point  D,  from  which 
C  and  A  are  visble,  and  a  point  E 
from  which  C  and  B  are  visible. 
At  C  observe  the  angles  DCA,ACB 
and  BCE.  Measure  the  distances  CD  and  CE.  Observe  the 
angles  ADC  and  BEC.  Then  in  the  triangle  ADC,  two  angles  and 
the  included  side  are  given,  to  find  CA  ;  and  the  same  in  the  tri- 
angle CBE,  to  find  CB.  Lastly,  in  the  triangle  ACB  "-wo  9idea 
and  the  included  angle  are  Viaown,  to  find  AB. 


292 


OBSTACLES  IIV  ANGULAR  SIRVEIING.      [part  vii 


1) 


(433)  Problem.  To  measure  an  inaccessible  distance,  AB, 
when  no  point  can  he  found  from  wliich  the  two  ends  can  he  seen 
Let  C  be  a  point  from  which  A  is 
visible,  and  D  a  point  from  which 
B  is  visible,  and  also  C.  Measure 
CD.  Find  the  distances  CA  and 
DB,  as  in  the  preceding  problem  ; 
i.  e.  choose  a  point  E,  from  which  A 
and  C  are  visible,  and  another 
point,  F,  from  which  D  and  B  are  visible.  Measure  EC  and  DF. 
Observe  the  angles  AEC,  ECA,  BDF  and  DFB  ;  and  at  the  same 
time  the  angles  ACD  and  CDB,  for  the  subsequent  work.  Then 
CA  and  DB  will  be  found,  as  were  CA  and  CB  in  the  last  problem. 
Then  in  the  triangle  CDB,  two  sides  and  the  included  angle  are 
known  to  find  CB  and  the  angle  DCB  ;  and,  lastly,  in  the  triangle 
ACB,  two  sides  and  the  included  angle  (the  difference  of  ACD 
and  DCB)  to  find  AB. 


(434)  Problem.    To  mterpolate  a  Base. 

objects,  A,  B,  C,  D,  heing  in  a  right 
line,  and  visible  from  only  one  point, 
E,  it  is  required  to  detervnine  the  dis- 
tance between  the  middle  points,  B 
and  C,  the  exterior  distances,  AB 
and  CD,  being  knoivn. 

LetAB  =  a,  CD  =  5,  BC  =  a:; 
AEB  =  P,  AEC  =  Q,  AED  =  R  . 

Calculate  an  auxiliary  angle,  K,  such  that 


Four  hiaccessihU 

Fig,  297. 
B    WW   '■^    C.  D 


tang.  2  K  = 


Aah 


sin. 


(R-P) 


{a  — by 


P  .sin.  (R— Q)' 


Then  is  a;  =  —  *——  ±  k— ?-• 

2  1  .  COS.  iv 

Of  the  two  values  of  x,  the  positive  one  is  alone  to  be  taken. 

This  problem  is  used  in  Triangular  Surveying  when  a  portion  of 
a  Base  line  passes  over  water,  &c. 


CHAP.  III.] 


Obstacles  to  Measurement, 


293 


(435)  Pi'oblenii      G-iven  the  angles  observed,  at  the  ends  of  a 

I'me  which  cannot  he  measured,  hetiveen  it  and  the  ends  of  a  line 

of  knoivn  length  hut  inaccessible,  required  the  length  of  the  former 

line.     This  Problem  is  the  converse  of  that  given  in  Art.  (430). 

Its  figure,  292,  may  represent  the  case,  if  the  distance  AB  be 

regarded  as  knoivn  and  CD  as  that  to  be  found.     Use  the  first  and 

Becond  formulas  as  before,  and  invert  the  last  formula,  obtaining 

^-r^        KTyS'm.  CBD  .  sin.  CAB 

CD  =  AB . 

sin.  BDC  .  sin.  ACB 

This  problem  may  also  be  solved,  indirectly,  by  assuming  any 
length  for  CD,  and  thence  calculating  as  in  the  first  part  of  Art. 
(430),  the  length  of  AB  on  this  hypothesis.  The  imaginary 
figure  thus  calculated  is  similar  to  the  true  one ;  and  the  true 
length  of  CD  will  be  given  by  this  proportion ;  calculated  length 
of  AB  :  true  length  of  AB  : :  assumed  length  of  CD  :  true  length 
of  CD. 

The  length  of  CD  can  also  be  obtained 
graphically.  Take  a  line  of  any  length, 
as  CD',  and  from  C  and  D'  lay  off  angles 
ecjual  to  those  observed  at  C  and  D,  and 
thus  fix  points  A,  B'.  Produce  AB'  till  it 
equals  the  given  distance  AB,  on  any  de- 
sired scale.  From  B  draw  a  parallel  to 
B'D',  meeting  AD'  produced  in  D ;  and 
from  D  draw  a  parallel  to  D'C  meeting 
AC  produced  in  C.  Then  CD  will  be  the 
the' same  scale  as  AB.*  * 


-^n 


Cf^:^ 


iquired  distance  to 


(436)  Problem.  Three  points,  A,  B,  C,  being  given  by  their 
distances  from  each  other,  and  two  other  points,  P  and  Q,  being 
so  situated  that  from  each  of  theyn  two  of  the  three  points  can  be 
teen  and  the  angles  APQ,  BPQ,  CQP,  BQP,  be  measured,  it  is 
required  to  determine  the  positions  of  P  ayid  Q. 


See  Article  (458)  for  a  solution  of  this  problem  by  the  Plane-Table. 


294  OBSTACLES  IN  ANGULAR  SURVEYIIVO.     [part  vii 

Construction.  Begin, 
as  in  Art.  (397),  hj  describ- 
ing a  circle  passing  through 
A  and  B,  and  having  the  cen- 
tral angle  subtended  by  AB, 
equal  to  twice  the  given  an-  p^-'— 
gle  APB.  and  thus  contain- 
ing that  angle.  The  point  '^v,^  '^v^^,-''  ,y' 
P  will  lie  somewhere  in  its  ' 
circumference.  Describe  another  circle  passing  through  B  and  C, 
and  having  a  central  angle  subtended  by  BC  equal  to  twice  the 
given  angle  BQC.  The  point  Q  will  lie  somewhere  in  its  :ir- 
cumference.  From  A  draw  a  line  making  with  AB  an  angle  = 
BPQ,  and  meeting  at  X  the  circle  first  drawn.  From  C  draw  a 
line  making  with  CB  an  angle  =  BQP,  and  meeting  the  second 
circle  in  Y.  Join  XY  and  produce  it  till  it  cuts  the  circles  in 
points  P  and  Q,  which  will  be  those  required  ;  since  BPX  =  BAX 
=  BPQ  ;  and  BQY  =  BCY  =  BQP. 

Calculation.  In  the  triangle  ABC,  the  sides  being  given,  the 
angle  ABC  is  known.  In  the  triangle  ABX,  a  side  and  all  the 
angles  are  known,  to  find  BX.  In  the  triangle  CBY,  BY  is  simi- 
larly found.  By  subtracting  the  angle  ABC  from  the  sum  of  the 
angles  ABX  and  CBY,  the  angle  XBY  can  be  obtained.  Then 
in  the  triangle  XBY,  the  sides  BX,  BY,  and  the  included  angle 
are  given  to  find  the  other  angles.  Then  in  the  triangle  BPX 
are  known  all  the  angles  and  the  side  BX  to  find  BP.  In 
the  triangle  BQY,  BQ  is  found  iif  like  manner.  Finally,  in  the 
triangle  BPQ,  PQ  can  then  be  found. 

If  desired,  we  can  also  obtain  AP  in  the  triangle  APB  ;  and  CQ 
in  the  triangle  CBQ. 

(437)  Problem.  Four  points,  A,  B,  C,  D,  being  given  in 
position,  hy  their  mutual  distances  and  directions,  and  tivo  other 
points,  P  and  Q,  being  so  situated  that  from  each  of  them  tivo  of 
the  four  points  can  be  seen  and  the  angles  APB,  APQ,  PQC  and 
PQD  measured,  it  is  required  to  determine  the  position  of  P  and  Q 


CKAP.  III.] 


Obstacles  to  Measurement. 


296 


Construction.  Begin  as  in  the  last  artic.e,  by  descrit)ing  on 
AB  the  segment  of  a  circle  to  contain  an  angle  equal  to  APB. 
From  B  draw  a  chord  BE,  making  an  angle  with  BA  equal  to  the 
supplement  of  the  angle  APQ.  On  CD  describe  another  segment 
to  contam  an  angle  equal  to  CQD.  From  C  draw  a  chord  CF, 
making  an  angle  with  CD  equal  to  the  supplement  of  the  angle 
DQP.  Draw  the  line  EF,  and  it  will  cut  the  two  circles  in  the 
required  points  P  and  Q.* 

Calculation.  To  obtam  PQ  =  EF  —  EP  —  QF,  we  proceed 
to  find  those  three  Imes  thus.  In  the  triangle  ABE,  we  know  the 
side  AB,  the  angle  ABE,  and  the  angle  AEB  =  APB  ;  whence 
to  find  EB.  In  the  same  way,  the  triangle  CFD  gives  FC.  In 
the  triangle  EBC  are  known  EB  and  BC,  and  the  angle  EBC  = 
ABC  —  ABE;  whence  EC  and  the  angle  ECB  are  found.  In 
the  triangle  ECF  are  kno^vn  EC,  FC,  and  the  angle  ECF  =  BCD 
—  ECB  — FCD;  whence  we  find  EF,  and  the  angles  CEF  and 
CFE. 

In  the  triangle  BEP,  we  have  EB,  the  angle  BEP  =  BEC  + 
CEP,  and  the  angle  BPE  =  BPA  +  APE  ;  to  find  EP  and  PB. 
In  the  triangle  QCF,  we  have  CF,  and  the  angles  CQF  and  CFQ, 
to  find  QC  and  QF.     Then  we  know  PQ  =  EF  —  EP  —  QF. 


•  For,  the  angle  APQ  in  the  figure  equals  the  measured  angle  APQ,  because 
the  supplement  of  the  former,  EPA,  equals  the  supplement  of  the  latter,  since  it 
is  measured  by  the  same  arc  as  the  angle  ABE,  equal  to  that  supplement  by  con- 
•tructioa.     So  too  with  the  angle  DQP. 


296 


OBSTACLES  L\  AXGULAR  SURVEYING,     [part  vn 


The  other  distances,  if  desired,  can  be  easily  found  from  the  above 
data,  some  of  the  calculations,  not  needed  for  PQ,  being  made  with 
reference  to  them.  In  the  triangle  ABP,  we  know  AB,  BP,  and 
the  angle  BAP,  to  find  the  angle  ABP  and  AP.  In  the  triangle 
QDC  we  know  QC,  CD,  and  the  angle  CQD,  to  find  the  angle 
QCD  and  QD.  In  the  triangle  PBC,  we  know  PB,  BC,  and  the 
angle  PBC  =  ABC  —  ABP,  to  find  PC.  Lastly,  in  the  triangle 
QCB,  we  know  QC,  CB,  and  the  angle  QCB  =  DCB  — DCQ,  to 
find  QB. 

The  solution  of  this  problem  includes  the  two  preceding ;  for,  let 
the  line  BC  be  reduced  to  a  point  so  that  its  two  ends  come  toge- 
ther and  the  three  lines  become  two,  and  we  have  the  problem 
of  Art.  (436)  ;  and  let  the  line  AB  be  reduced  to  a  point,  B, 
and  CD  to  a  point,  C,  and  we  have  but  one  line,  and  the  problem 
becomes  that  of  Art.  (435). 

In  these  three  problems,  if  the  two  stations  lie  in  a  right  line 
with  one  of  the  given  points,  the  problem  is  indeterminate. 


Four  points,  A,  B,  C,  D, 

Fisr.  301. 


B 


(438)  Problem  of  the  eight  points 

are  inaccessible,  hut  visible 
from  four  other  points,  E, 
F,  G,  H  ;  it  is  required  to  . 
find  the  respective  distances  '*• 
of  these  eight  points ;  the 
only  data  being  the  obser- 
vation, from  each  of  the 
points  of  the  second  sys- 
tem, of  the  angles  under 
which  are  seen  the  points 
vj  the  first  system. 


This  problem  can  be  solved,  but  the  great  length  and  complicar 
tion  of  the  investigation  and  resulting  formulas  render  it  more  a 
matter  of  curiosity  than  of  utility.  It  may  be  found  in  Puissant's 
"  Topographic,"  page  55  ;  Lefevre's  "  Trigonometric"  p.  90,  and 
Lefevre's  '•^  Arp>entage,"  No.  387. 


-< 


OHAP,  IV.] 


To  Supply  Omissions 


297 


^ 


CHAPTER  IV. 


TO  SUPPLY  omssioxs. 


(439)  Any  two  omissions  in  a  closed  survey,  whether  of  the 
direction  or  of  the  length,  or  of  both,  of  one  or  more  of  the  sides 
bounding  the  area  surveyed,  can  always  be  supplied  by  a  suitable 
application  of  the  principle  of  Latitudes  and  Departures,  as  was 
stated  in  Art.  (283)  ;  although  this  means  should  be  resorted  to 
only  in  cases  of  absolute  necessity,  since  any  omission  renders  it 
impossible  to  "Test  the  survey,"  as  directed  in  Art.  (282).  In 
the  following  articles  the  survey  will  be  considered  to  have  been 
made  with  the  Compass.  All  the  rules  will  however  apply  to  a 
Transit  or  Theodolite  survey,  the  angles  being  referred  to  any  hne 
as  a  meridian,  as  in  "  Traversing." 

To  save  unnecessary  labor,  the  examples  in  the  various  cases 
now  to  be  examined,  will  all  be  taken 
from  the  same  survey,  a  plat  of  which 
is  given  in  the  margin  on  the  scale  of 
40  chains  to  1  inch  (1:31,680),  and 
the  Field-notes  of  which,  with  the 
Latitudes  and  Departures  carried  out  a 
to  five  decimal  places,  are  given  on 
the  following  page.* 


*  The  teacher  can  make  any  number  of  examples  for  his  own  use  hy  taking 
a  tolerably  accurate  survey,  striking  out  the  bearing  and  distance  of  any  one 
pourse,  and  calculating  it  precisely  as  in  Case  1,  given  below.  He  can  then  omit 
any  Iwo  quantities  at  will^  to  be  supplied  by  the  student  by  means  of  the  rules 
nov/  to  be  given. 


298 


OBSTACLES  IN  ANGULiR  SURVEriNG.     [part  vii. 


BEARING. 

DIST. 
IN   LINKS. 

LATITUDKS. 

DEPARTURES.               | 

N. 

f-',. 

E. 

W. 

A 

North. 

1284 

1284.00000 

0 

°    i 

B 

N.  3'J°  E. 

1782 

1511.22171 

944.31619 

1 

0 

N.  80°  E. 

2400 

41C.75.568 

2363.53872 

D 

S.  48*^  E. 

2700 

1806.G52r2 

2006.49096 

. 

E 

S.  18°  W. 

2860 

2720.02159 

883.78862 

F 

N.  73°  28' 21"  W. 

462  li 

1314.G9682 

4430.55725, 

4o26.67421    4526.67421    5314.34587  5314.34587 

Case  1.  When  the  length  and  the  Bearing  of  any  one  side  are 
wanting. 

(440)  Find  the  Latitudes  and  the  Departures  of  the  remaining 
sides.  The  difference  of  the  North  and  South  Latitudes  of  these 
lines,  is  the  Latitude  of  the  omitted  hne,  and  the  difference  of  their 
Departiu-es  is  its  Departure.  This  Latitude  and  Departure  are  two 
sides  of  a  right  angled  triangle  of  which  the  omitted  line  is  the 
hjpothenuse..  Its  length  is  therefore  equal  to  the  square  root  of 
the  sum  of  their  squares,  and  the  quotient  of  the  Departure  divided 
bj  the  Latitude  is  the  tangent  of  its  Bearing ;  as  in  Art.  (417). 

In  the  above  survey,  suppose  the  course  from  F  to  A  to  have 
been  omitted  or  lost.  The  difference  of  the  Latitudes  of  the 
remaining  courses  will  be  found  to  be  1314.69682,  and  the  differ- 
ence of  the  Departures  to  be  4430.5572.5.  The  square  root  of  the 
sum  of  their  squares  is  4621.5  ;  and  the  quotient  of  the  Departure 
divided  by  the  Latitude  is  the  tangent  of  73°  28'  21".  The  de- 
ficiencies were  in  North  Latitude  and  West  Departure ;  and  the 
omitted  course  is  therefore  N.  73°  28'  21"  W.,  4621.5 

Case  2.   When  the   length  of  one  side  and  the  Bearing  of 
\  '    another  are  wanting, 

(441)  When  the  deOcient  sides  adjoin  each  other.  Find,  as 
in  Case  1,  the  length  and  Bearing  of  the  line  joining  the  ends  of 
the  remaining  courses.  This  line  and  the  deficient  hnes  ^vill  form  a 
triangle,  in  which  two  sides  will  be  known,  and  the  angle  between 
the  calculated  side  and  the  side  whose  Bearing  is  given  can  be 
found  by  Art.  (243).  The  parts  wanting  can  then  be  obtained 
by  the  common  rules  of  Trigonometry. 


CUkP.  IV. 


To  Supply  Omissions. 


299 


In  the  figure,  let  the  length  of  EF, 
and  the  Bearing  of  FA  be  the  omitted 
parts.  The  difference  of  the  sums  of 
the  N.  and  S.  Latitudes,  and  the  E. 
and  W., Departures  of  the  complete  B| 
courses  from  A  to  E,  are  respectively 
1405.32477  North  Latitude,  and  -^" 
5314.34587  East  Departure.  The 
course,  EA,  corresponding  to  this  de- 
Sciency  we  find,  by  proceedmg  as  in  case  1,  to  be  S.  75°  11'  15 
W.,  5497.026.  The  angle  AEF  is  therefore  =  75°  11'  15"  - 
18°  =  57°  11'  15".  Then  in  the  triangle  AEF  are  given  the 
sides  AE,  AF,  and  the  angle  AEF  to  find  the  remainmg  parts ; 
viz.  the  angle  AFE  =  91°  28'  21",  whence  the  Bearmg  of 
FA  =  91°  28'21"  — 18°  =  N.  73°  28'  21"  W.;  and  the  side 
EF  =  2860. 


(142)  When  the  deficient  sides  are  separated  from  each  other. 

A  modification  of  the  preceding  method  will  still  apply.  Li  thia 
figure  let  the  omissions  be  the  Bearing 
of  FA  and  the  length  of  CD.  Imagine 
the  courses  to  change  places  without 
changing  Bearings  or  lengths,  so  as  to 
bring  the  deficient  lines  next  to  each 
other,  by  transferring  CD  to  AG,  AB 
to  GH,  and  BC  to  HD.  This  wiU  not 
affect  their  Latitudes  or  Departures. 
Join  GF.  Then  in  the  figure  DEFGH, 
the  Latitudes  and  Departures  of  all  the  sides  but  FG  are  known, 
whence'  its  length  and  Bearing  can  be  found  as  in  Case  1. 
Then  the  triangle  AGF  may  be  treated  like  the  triangle  AEF 
in  the  last  article,  to  obtain  the  length  of  AG  =  CD,  and  the  Bear- 
mg of  FA. ' 


(443)  Otherwise^  hy  clianging  the  3Ieridian.  Imagine  the  field 
to  turn  around,  till  the  side  of  which  the  distance  is  unknown, 
becomes  the  Meridian,  i.  e.  comes  to  be  due  North  and  South  * 


300  OBSTACLES  IN  AXGUAR  SIRVEYLXG.      [part  vii 

all  the  other  sides  retaining  their  relative  positions,  and  continuing 
to  make  the  same  angles  with  each  other.  Change  their  Bearings, 
accordingly,  as  directed  ui  Art.  (244).  Find  the  Latitudes  and 
Departures  of  the  sides  in  their  new  positions.  Since  the  side 
whose  length  was  unknovna  has  been  made  the  Meridian,  it  has  no 
Departure,  whatever  may  be  its  unknown  length ;  and  the  difference 
of  the  columns  of  Departure  will  therefore  be  the  Departure  of  the 
side  whose  Bearing  is  unknown.  The  length  of  this  side  is  given. 
It  is  the  hypothenuse  of  a  right  angled  triangle,  of  which  the  De- 
parture is  one  side.  Hence  the  other  side,  which  is  the  Latitude, 
can  be  at  once  found ;  and  also  the  unknown  Bearing. 

Put  this  Latitude  in  the  Table  in  the  blank  where  it  belonss. 
Then  add  up  the  columns  of  Latitude,  and  the  difference  of  their 
sums  will  be  the  unknown  length  of  the  side  which  had  been  made 
a  Meridian.* 

Let  the  omitted  quantities  be,  ^as  in  the  last  article,  the  length 

of  CD  and  the  [Bearing  of  FA. 
Make  CD  the  Meridian.  The  chang- 
ed Bearings  will  then  be  found  by 
Art.  (244)  to  be  as  in  the  margin. 
To  aid  the  imagination,  turn  the 
book  around  till  CD  points  up  and 
down,  as  North  lines  are  usually 
placed  on  a  map.  Then  obtain  the  Latitudes  of  the  courses  with 
their  new  Bearings  and  old  distances,  and  proceed  as  has  been 
directed. 

Case  3.      When  the  lengths  of  two  sides  are  wanting. 

(444)  When  the  deScient  sides  a^oin  each  other.  Find  the 
Latitudes  and  Departures  of  the  other  courses,  and  then,  by  Case 
1 ,  find  the  length  and  Bearing  of  the  line  joining  the  extremities 
of  the  deficient  courses.  Then,  in  the  triangle  thus  formed,  are 
known  one  side  and  all  the  angles  (deduced  from  the  Bearings)  to 
find  the  lengths  of  the  other  two  sides. 

•  This  conception  of  thus  changing  the  Bearings  is  stated  to  be  due  to  Pro; 
Robert  Patterson,  of  Philadelphia,  by  whom  it  was  communicated  to  Mr  John 
Gammere,  and  published  by  him,  in  1814,  in  his  "  Treatise  on  Surveying  " 


STA. 

OLD    BEARING. 

NEW    BEARINfJ. 

A 

North. 

N.  8U°  W. 

B 

i\.  32°  E. 

N.  48°  W. 

C 

N.  80°  E. 

Norlk. 

D 

S.  48°  E. 

N.  52°  E. 

E 

S.  18°  W. 

S.  6^'=  E. 

F 

CHAP.  TV.]  fo  Snpply  Omissions.  301 

Thus,  in  Fig.  303,  page  299,  let  EF  and  FA  be  the  sides  whose 
.engths  are  unkno-vvn.  EA  is  then  to  be  calculated,  and  its  length 
will  be  found,  as  in  Art.  (HI),  to  be  5497.026,  and  its  bearmg 
S.  75°  11'  15"  W.,  whence  the  angle  AEF  =  75''  11'  15"  — 18' 
=  57°  11'  15" ;  AFE  =  18°  +  73°  28'  21"  =  91°  28'  21" ;  and 
E AF  =  31°  20'  24" ;  whence  can  be  obtamed  EF  =  2860  and 
FA  =  4621.5. 

(445)  When  the  deficient  sides  are  separated  from  each  otlier 

Let  the  lengths  of  BC  and  DE  be  those 
omitted.  Again  imagine  the  courses 
to  change  places,  so  as  to  bring  the 
deficient  lines  together,  DE  being 
transferred  to  CG,  and  CD  to  GE. 
Join  BG.  Then  in  the  figure 
ABGEFA,  are  known  the  Latitudes 
and  Departures  of  all  the  courses  ex- 
cept BG,  whence  its  length  and  Bearing 
can  be  found  as  in  Case  1.  Then  in  the  triangle  BCG,  the  angle 
CBG  can  be  found  from  the  Bearings  of  CB  and  BG,  and  the  angle 
CGB  from  the  Bearings  of  BG  and  GC.  Then  all  the  angles  of 
tlie  triangle  are  kno^vn  and  one  side,  BG,  whence  to  find  the 
required  sides,  BC  =  1782,  and  CG  =  DE  =  2700. 

(446)  Otherwise,  hy  changing  the  Meridian.  As  in  Art.  (443), 
imagine  the  field  to  turn  around,  till  one  of  the  sides  whose  length 
is  wanting,  becomes  a  Meridian  or  due  North  and  South.  Change 
all  the  Bearings  correspondingly.  Find  the  Latitudes  and  Depar- 
tures of  the  changed  courses.  The  difference  of  the  columns  of 
Departure  will  be  the  Departure  of  the  second  course  of  unknown 
length,  since  the  course  made  Meridian  has  now  no  Departure. 
The  new  Bearing  of  this  second  course  being  given,  in  the  right 
angled  triangle  formed  by  this  course  (as  an  hypothenuse)  and  it3 
Departure  and  Latitude,  we  know  one  side,  the  Departure,  and 
the  acute  angles,  which  are  the  Bearing  and  its  complement.  The 
length  of  the  course  is  then  readily  calculated  ;  and  also  its  Lati- 
xude.     This  Latitude  being  inserted  in  its  proper  place,  the  iifFei> 


S02 


OBSTACLES  L\  AIVGULAR  SURVEYIIVG.     [part  vii. 


STA. 

A 

OLD    BEARING. 

NEW     BEARING. 

North. 

N.  32°   W. 

B 

N.  32°  E. 

North. 

C 

N.  80°  B. 

N.  48°  E. 

D 

S   48°  E. 

S.  80°  E. 

E 

s.  :s°  W. 

S.  14°  E. 

F 

N.  7.3°  28'  21"  W 

S.  74°  31'  39"  W. 

ence  of  the  columns  of  Latitude  will  be  the  length  of  that  wanting 
side  which  had  been  made  a  Meridian. 

Thus,  let  the  lengths  of  BC  and  DE  be  wanting,  as  in  the  pre- 
ceding example.  Make  BC 
a  Meridian.  The  other  Bear- 
ings are  then  changed  as  in 
the  margin.  Calculate  new 
Latitudes  and  Departures. 
The  difference  of  the  Depar* 
tures  will  be  the  Departure 
of  DE,  since  BC,  being  a  Meridian,  has  no  Departure.  Hence  the 
length  and  Latitude  of  DE  are  readily  obtained.  This  Latitude 
being  put  in  the  table,  and  the  columns  of  Latitude  then  added  un. 
their  difference  will  be  the  length  of  BC. 

Case  4.      When  the  Bearings  of  two  sides  are  wanting. 

(147)  \l  hen  the  deflcient  sides  adjoin  each  other.  Find  the 
Latitudes  and  Departures  of  the  other  sides,  and  then,  as  in  Case 
1,  find  the  length  and  bearing  of  the  line  joining  the  extremities 
of  the  deficient  sides.  Then  in  the  triangle  thus  formed  we  have 
the  three  sides  to  find  the  angles  and  thence  the  Bearings. 


(418)  When  the  deficient  sides  are  separated  from  each  other 

Change  the  places  of  the  sides  so  as  to  bring  the  deficient  ones 

next  to   each   other.      Thus,  in   the  Fig.  306. 

figure,  supposing  the  Bearings  of  CD, 

and  EF  to  be  wanting,  transfer  EF  to 

DG,  and  DE  to  GF.     Then  calculate, 

as  in  Case  1,  the  length  and  Bearing 

of  the  line  joining  the  extremities  of 

the  deficient  sides,  CG  in  the  figure. 

This  line  and  the  deficient  sides  form  a 

triangle  in  which  the  three  sides  are 

given  to  determine  the  angles  and  thence  the  required  Bearmgs. 

*  The  fullest  investigation  of  this  subject,  developing  many  curious  points,  will 
be  found  in  Mascheroni's  "Pmhlemes  de  Oeometrie  pour  les  Arpenteurs,"  and  Lhu 
tJlier's  ''Folygonomet'ie."    The  method  of  Arts.  (442),  (445),  and  (448)  is  new. 


PAET  viir. 


PLANE  TABLE  SURVEYING, 


(IW)  TiiE  Plane  Table  is  in  substance  merely  a  drawing  board 
fixed  on  a  tripod,  so  that  lines  may  be  drawn  on  it  by  a  ruler  placed 
BO  as  to  point  to  any  object  in  sight.  All  its  parts  are  mere  addi- 
tions to  render  this  operation  more  convenient  and  precise.* 

Such  an  arrangement  may  be  applied  to  any  kind  of  "  Angular 
Sui'veying" ;  such  as  the  Thu'd  Method,  "  Polar  Surveying,"  in 
its  two  modifications  of  Radiation  and  Progression^  (characterized 
in  Art.  (220)),  and  the  Fourth  Method,  by  Intersections.  Each 
of  these  will  be  successively  explained.  The  instrument  is  very 
convenient  for  filhng  in  the  details  of  a  survey,  when  the  principal 
points  have  been  determined  by  the  more  precise  method  of  "  Tri- 
angular Survej^ing,"  and  can  then  be  platted  on  the  paper  in 
advance.  It  has  the  great  advantage  of  dispensing  with  all 
notes  and  records  of  the  measurements,  since  they  are  platted  aa 
tliey  are  made.  It  thus  saves  time  and  lessens  mistakes,  but  is 
wanting  in  precision. 

(450)  The  Table.  It  is  usually  a  rectangular  board  of  well 
seasoned  pine,  about  20  inches  wide  and  30  long.  The  paper  to 
be  drawn  upon  may  be  attached  to  it  by  drawing-pins,  or  by  clamp- 
ing plates  fixed  on  its  sides  for  that  purpose,  or  by  springs  pressed 
upon  it,  or  it  may  be  held  between  rollers  at  opposite  sides  of  the 
table.  Tmted  paper  is  less  dazzling  m  the  sun.  Cugnot's  jomt, 
described  on  page  134,  is  the  best  for  connecting  it  with  its  tripod, 
tliough  a  pair  of  parallel  plates,  like  those  of  the  Theodohte,  are 
often  used.  A  detached  level  is  placed  on  the  board  to  test  its 
horizontality  ;  though  a  smooth  ball,  as  a  marble,  will  answer  the 
same  purpose  approximately. 

•  The  Plane  Table  is  not  a  Goniometer,  or  Angle-  measure,  Mke  the  Compas^  Tnutdt,  te. 
but  ft  Oonigraph,  or  Angl&-drav)er, 


304  PLANE  TABLE  SURVEYOG.  [pakt  viii. 

A  pair  of  sights,  like  those  of  the  compass,  are  sometimes 
placed  under  the  board,  serving,  like  a  "  Watch  Telescope,"  (Art. 
(339),  to  detect  any  movement  of  the  instrument.  To  find  what 
point  on  the  lower  side  of  the  board  is  exactly  under  a  point  on 
the  upper  side,  so  that  by  suspending  a  plumb-hne  from  the  former 
the  latter  may  be  exactly  over  any  desired  point  of  ground,  a  large 
pair  of  "  callipers,"  or  dividers  with  curved  legs,  may  be  used,  one 
of  their  points  being  placed  on  the  upper  point  of  the  board,  and 
their  other  point  then  determining  the  corresponding  under  point ; 
or  a  frame  forming  three  sides  of  a  rectangle,  like  a  slate  frame, 
may  be  placed  so  that  one  end  of  one  side  of  it  touches  the  upper 
point,  and  the  end  of  the  corresponding  side  is  under  the  table 
precisely  below  the  given  point,  so  that  from  this  end  a  plumb-line 
can  be  dropped.  A  compass  is  sometimes  attached  to  the  table, 
or  a  detached  compass,  consisting  of  a  needle  in  a  narrow  box, 
(called  a  Declinator),  is  placed  upon  it,  as  desired.  The  edges 
of  the  table  are  sometimes  divided  into  degrees,  hke  the  "  Drawing 
board  Protractor,"  Art.  (273).  It  then  becomes  a  sort  of  Gonio 
meter,  hke  that  of  Art.  (213). 

(451)  The  Alidade*  The  ruler  has  a  fiducial  or  feather  edge, 
which  may  be  divided  into  inches,  tenths,  &c.  At  each  end  it 
carries  a  sight  hke  those  of  the  compass.  Two  needles  would  be 
tolerable  substitutes.  The  sights  project  beyond  its  edge  so  that 
their  centre  lines  shall  be  precisely  in  the  same  vertical  plane  as  this 
e  Ige,  in  order  that  the  lines  drawn  by  it  may  correspond  to  the 
lines  sighted  on  by  them.  To  test  this,  fix  a  needle  in  the  board, 
piace  the  ruler  against  it,  sight  to  some  near  point,  draw  a  fine 
by  the  ruler,  turn  it  end  for  end,  again  place  it  against  the  needle, 
again  sight  to  the  same  point,  and  draw  a  new  line.  If  it  coincides 
with  the  former  line,  the  above  condition  is  satisfied.  The  ruler 
and  sights  together  take  the  name  of  Alidade.  If  a  point  should 
be  too  high  or  too  low  to  be  seen  with  the  ahdade,  a  plumb-line, 
held  between  the  eye  and  the  object,  will  remove  the  difficulty. 

A  telescope  is  sometimes  substituted  for  the  sights,  being  sup- 
ported above  the  ruler  by  a  standard,  and  capable  of  pointing 
upward  or  downward.     It  admits  of  adjustments  similar  in  principle 


PART  VIII.] 


PLAXE  TABLE  SURVETIXG. 


305 


to  the  2d  and  3d  adjustments  of  the  Transit,  Part  IV,  Chapter  3, 
pages  242  and  246. 

But  even  without  these  adjustments,  whether  of  the  sights  or 
of  the  telescope,  a  survey  could  be  made  which  would  be  per- 
fectly correct  as  to  the  relative  position  of  its  parts,  however  far 
the  line  of  sight  might  be  from  lying  in  the  same  vertical  plane 
as  the  edge  of  the  ruler,  or  even  from  being  parallel  to  it ;  just  aa 
in  the  Transit  or  Theodohte  the  index  or  vernier  need  not  to  be 
exactly  under  the  vertical  hair  of  the  telescope,  since  the  angular 
deviation  affects  all  the  observed  directions  equally. 


(452)  Method  of  Radiation.  This  is  the  simplest,  though  not 
the  best,  method  of  surveying  with  the  Plane-table.  It  is  especi- 
ally applicable  to  survey- 
ing a  field,  as  in  the  figure. 
In  it  and  the  following  fi- 
gures, the  size  of  the  Table 
is  much  exaggerated.  Set 
the  instrument  at  any  conve- 
nient point,  as  0 ;  level  it, 
and  fix  a  needle  (having  a 
head  of  seahng-wax)  in  the 
board  to  represent  the  sta- 
tion. Direct  the  alidade  to  any  comer  of  the  field,  aa  A,  the  fiducial 
edge  of  the  ruler  touching  the  needle,  and  draw  an  indefinite  line  by 
it.  Measure  OA,  and  set  off  the  distance,  to  any  desired  scale,  from 
the  needle  point,  along  the  line  just  drawn,  to  a.  The  fine  OA  is 
thus  platted  on  the  paper  of  the  table  as  soon  as  determined  in  the 
field.  Determine  and  plat  in  the  same  way,  OB,  OC,  &c.,  to  b,  (?, 
&c.  Join  ab,  be,  &c.,  and  a  complete  plat  of  the  field  is  obtained. 
Trees,  houses,  hills,  bends  of  rivers,  &c.,  may  be  determnied  in 
the  same  manner.  The  corresponding  method  with  the  Compass 
or  Transit,  was  described  hi  Articles  (258)  and  (391).  The  table 
may  be  set  at  one  of  the  angles  of  the  field,  if  more  convenient. 
If  the  alidade  has  a  telescope,  the  method  of  measuiing  distances 
with  a  stadia,  described  m  Art.  (375),  may  be  here  applied  with 
great  advantage. 

20  . 


306 


PLANE  TABLE  SIIRVEYL\G. 


[part  Tin 


c 

E 

\ 

i     \ 

^L  y^ 

^^ 

(453)  Method  of  Progression.  Let  ABCD,  &c.,  be  the  line 
to    be    surveyed.  ^^s-  308. 

Fix  a  needle  at  a 
convenient  point 
of  the  Plane-table, 
near  a  corner  so 
as  to  leave  room 
for  the  plat,  and 
set  up  the  table  at 
B,  the  second  an- 
gle of  the  line,  so 
that  the  needle, 
whose  point  repre- 
sents B,  and  which  should  be  named  5,  shall  be  exactly  over  that 
station.  Sight  to  A,  pressing  the  fiducial  edge  of  the  ruler  against 
the  needle,  and  draw  a  line  by  it.  Measure  BA,  and  set  off  its 
length,  to  the  desired  scale,  on  the  line  just  drawn,  from  6  to  a 
pomt  a,  representing  A.  Then  sight  to  C,  draw  an  indefinite  line 
by  the  ruler,  and  on  it  set  off  the  length  of  BC  from  h  to  c.  Fix 
the  needle  at  c.  Set  up  at  C,  the  point  c  being  over  this  station, 
and  make  the  line  cb  of  the  plat  coincide  in  direction  with  CB  on 
the  ground,  by  placing  the  edge  of  the  ruler  on  cb,  and  turning  the 
table  till  the  sights  point  to  B.  The  compass,  if  the  table  have 
one,  will  facilitate  this.  Then  sight  forward  from  C  to  D,  and  fix 
CD,  gd  on  the  plat,  as  be  was  fixed.  Set  up  at  D,  make  do  coincide 
with  DC,  and  proceed  as  before.  The  figure  shews  the  lines 
drawn  at  each  successive  station.  The  Table  drawn  at  A  shews 
how  the  survey  might  be  commenced  there. 

In  going  around  a  field,  the  work  would  be  proved  by  the  last 
line  "  closing"  at  the  starting  point ;  and,  durmg  the  progress  of 
the  survey,  by  any  direction,  as  from  C  to  A  on  the  ground,  coin- 
ciding with  the  corresponding  line,  ca,  on  the  plat. 

This  method  is  substantially  the  same  as  the  method  cf  survey- 
ing a  line  with  the  Transit,  explained  in  Art.  (372).  It  requires 
all  the  points  to  be  accessible.  It  is  especially  suited  to  the  sur 
vey  of  a  road,  a  brook,  a  winding  path  through  woods,  &c.  The 
offsets  required  may  often  be  sketched  in  by  eye  with  sufiicient 
precision. 


PAKT  VIII.] 


PLA\E  TABLE  SCRVEYIXti. 


307 


When  the  paper  is  filled,  put  on  a  new  sheet,  and  begin  by  fixing 
on  it  two  points,  such  as  C  and  D,  which  were  on  the  former  sheet, 
and  from  them  proceed  as  before.  The  sheets  can  then  be  after- 
wards united,  so  that  all  the  points  on  both  shall  be  in  their  true 
relative  positions. 

(454)  Method  of  Intersection.  This  is  the  most  usual  and 
the  Buost  rapid  method  of  using  the  Plane-table.  The  principle 
was  referred  to  m  Articles  (259)  and  (392).  Set  up  the  instru- 
ment at  any  convenient  point,  as  X  in  the  figure,  and  sight  to  all 

Fig.  309. 
B  C 


aC- 


X  X 

the  desired  points  A,  B,  C,  &c.,  which  are  visible,  and  draw  inde- 
finite lines  in  their  directions.  Measure  any  line  XY,  Y  being 
one  of  the  points  sighted  to,  and  set  oflf  this  Hue  on  the  paper  to 
any  scale.  Set  up  at  Y,  and  turn  the  table  tiU  the  line  XY  on 
the  paper  lies  in  the  direction  of  XY,  on  the  ground,  as  at  C  in  the 
last  method.  Sight  to  all  the  former  points  and  draw  lines  in  their 
directions,  and  the  intersections  of  the  two  lines  of  sight  to  each 
point  wUl  determine  them,  by  the  Fourth  Method,  Art.  (8). 
Points  on  the  other  side  of  the  line  XY  could  be  determined  at  the 
same  time.  In  surveying  a  field,  one  side  of  it  may  be  taken  for 
the  base  XY.  Very  acute  or  obtuse  intersections  should  be 
avoided.  30°  and  150°  should  be  the  extreme  Umits.  The  impos- 
sibility of  always  doing  this,  renders  this  method  often  deficient  in 
precision.  ^Mien  the  paper  is  filled,  put  on  a  new  sheet,  by  fixing 
on  it  two  known  pomts,  as  m  the  preceding  method. 


S08 


PLAXE  TABLE  SURVEYING. 


[PAR^  VIII 


(455)  Method  of  Resection.    This  method  (called  by  the  French 
Recoupement)  is  a  modification  of  the  preceding  method  of  Inter 


Fig.  310. 


section.  It  requires  the  measurement  of  only  one  distance,  but  all 
the  points  must  be  accessible.  Let  AB  be  the  measured  distance. 
Lay  it  off  on  the  paper  as  ah.  Set  the  table  up  at  B,  and  turn  it 
till  the  line  ba  on  the  paper  coincides  with  BA  on  the  ground,  as 
in  the  Method  of  Progression.  Then  sight  to  C,  and  draw  an  inde- 
finite line  by  the  ruler.  .Set  up  at  C,  and  turn  the  line  last  drawn 
so  as  to  point  to  B.  Fix  a  needle  at  a  on  the  table,  place  the 
alidade  against  the  needle  and  turn  it  till  it  sights  to  A.  Then  the 
point  in  which  the  edge  of  the  ruler  cuts  the  line  drawn  from  B 
will  be  the  point  c  on  the  table.  Next  sight  to  D,  and  draw  an 
indefinite  line.  Set  up  at  D,  and  make  the  line  last  drawn  point 
to  C.  Then  fix  the  needle  at  a  or  b,  and  by  the  alidade,  as  at  the 
last  station,  get  a  new  fine  back  from  either  of  them,  to  cut  the  last 
drawn  line  at  a  point  which  will  be  d.  So  proceed  as  far  as  de- 
sired. \ 

(456)  To  orient  the  table.*  The  operation  of  orientation  con- 
sists in  placing  the  table  at  any  point  so  that  its  Hnes  shall  have 
the  same  directions  as  when  it  was  at  previous  stations  in  the  same 
survey. 

'  The  French  phrase,  To  orient  one's  self,  meaning  to  determine  one's  position, 
nsnally  with  respect  to  the  four  quarters  of  the  heavens,  of  which  the  Orient  is 
iie  leading  one,  well  deserves  naturalization  in  our  language. 


PART  VIII.] 


PLAAE  lABLE  SURFEriAG. 


309 


With  a  compass,  this  is  very  easily  effected  by  turning  the  table 
till  the  needle  of  the  attached  compass,  or  that  of  the  Declinator, 
placed  in  a  fixed  position,  pomts  to  the  same  degree  as  when  at 
the  pre\aous  station. 

Without  a  compass  the  table  is  oriented,  when  set  at  one  end  of 
a  Ime  previously  determined,  by  sightmg  back  on  this  Ime,  as  at  C 
m  the  Method  of  Progression,  Art.  (453). 

To  orient  the  table,  when  at  a  station  imconnected  with  others, 
is  more  difficult.  It  may  be 
effected  thus.  Let  ah  on  the  ta- 
ble represent  a  line  AB  on  the 
ground.  Set  up  at  A,  make  ah 
coincide  with  AB,  and  draw  a 
line  from  a  dii-ected  towards  a 
steeple,  or  other  conspicuous  ob- 
ject, as  S.  Do  the  same  at  B.  Draw  a  line  cd,  parallel  to  ah, 
and  mtercepted  between  aS,  and  5S.  Divide  ah  and  cd  into  the 
same  number  of  equal  parts.  The  table  is  then  prepared.  Now 
let  there  be  a  station,  P,  p  on  the  table,  at  which  the  table  is  to  be 
onented.  Set  the  table,  so  thatp  is  over  P,  apply  the  edge  of  the 
ruler  to  p,  and  turn  it  till  this  edge  cuts  cd  in  the  division  corre- 
sponding to  that  in  which  it  cuts  ah.  Then  turn  the  table  till  the 
sights  point  to  S,  and  the  table  will  be  oriented. 

(4j7)  To  Gud  cue's  place  ou  the  groaud.  This  problem  may 
be  otherwise  expressed  as  Interpolating  a  point  in  a  plat.  It  is 
most  easily  performed  by  reversing  the  Method  of  Intersection. 
Set  up  the  table  over  the  station, 
0  in  the  figure,  whose  place  on 
the  plat  already  on  the  table  is 
desired,  and  orient  it,  by  one  of 
the  means  described  in  the  last 
article.  Make  the  edge  of  the 
ruler  pass  through  some  point,  a 
on  the  table,  and  turn  it  till  the 
sights  point  to  the  corresponding 
station,  A  on  the  ground.     Draw  a  line  by  the  ruler.     The  desired 


310  PLAXE  TABLE  SURVEFING.  [part  nii 

point  is  somewhere  in  this  line.  Make  the  ruler  pass  through 
another  point,  h  on  the  table,  and  make  the  sights  point  to  B  od 
the  ground  Draw  a  second  line,  and  its  intersection  with  the 
first  will  be  the  pomt  desired.  Usuig  C  in  the  same  way  would 
give  a  third  line  to  prove  the  work.  This  operation  may  be  used 
as  a  new  method  of  surveying  with  the  plane-table,  since  any 
number  of  points  can  have  their  places  fixed  in  the  same  manner. 

This  problem  may  also  be  executed  on  the  principle  of  Trilinear 
Surveying.  Thi'ee  pomts  being  given  on  the  table,  lay  on  it  a  piece 
of  transparent  paper,  fix  a  needle  any  where  on  this,  and  with  the 
ahdade  sight  and  draw  lines  towards  each  of  these  three  points 
on  the  ground.  Then  use  this  paper  to  find  the  desired  point,  pre- 
cisely as  directed  in  the  last  sentence  of  Art.  (398),  page  277. 

(158)  Inaccessible  distancest  Many  of  the  problems  in  Part 
VII.  can  be  at  once  solved  on  the  ground  by  the  plane-table,  since 
it  is  at  the  same  time  a  Goniometer  and  a  Protractor.  Thus,  the 
Problem  of  Art.  (435)  may  be  solved  as  follows,  on  the  principle 
of  the  construction  in  the  last  paragraph  of  that  article.  Set  the 
table  at  C.  Mark  on  it  a  point,  c',  to  represent  C,  placing  c' 
vertically  over  C.  Sight  to  A,  B  and  D,  and  draw  corresponding 
lines  from  c'.  Set  up  at  D,  mark  any  point  on  the  fine  drawn  from 
c'  towards  D,  and  call  it  d'.  Let  d'  be  exactly  over  D,  and  direct 
d'c'  toward  C.  Then  sight  to  A  and  B,  and  draw  corresponding 
lines,  and  their  intersections  with  the  lines  before  drawn  towards 
A  and  B  will  fix  points  a'  and  h'.  Then  on  the  line  joining  a  and 
6,  given  on  the  paper  to  represent  A  and  B,  ah  being  equal  to  AB 
on  any  scale,  construct  a  figure,  abed,  similar  to  alh'ci'd',  and  the 
Jine  od  thus  determined  will  represent  CD  on  the  same  scale  as  AB 


PART  IX. 


bURVEYING  WITHOUT  INSTRUMENTS. 

(459)  The  Principles  which  were  estabUshed  in  Part  I,  and  subse- 
quently applied  to  surveying  with  various  instruments,  may  also  be 
employed,  with  tolerable  correctness,  for  determining  and  represent- 
ing the  relative  positions  of  larger  or  smaller  portions  of  the  earth's 
surface  without  any  Instruments  but  such  as  can  be  extemporized. 
The  prominent  objects  on  the  ground,  such  as  houses,  trees,  the 
summits  of  hills,  the  bends  of  rivers,  the  crossings  of  roads,  &c., 
are  regarded  as  "points"  to  be  "determined."  Distances  and 
angles  are  consequently  required.  Approximate  methods  of 
obtaining  these  will  therefore  be  first  given. 

(460)  Distances  Dy  pacing.  Quite  an  accurate  measurement 
of  a  line  of  ground  may  be  made  by  walking  over  it  at  a  uniform 
pace,  and  counting  the  steps  taken.  But  the  art  of  walking  in  a 
straight  Hne  must  first  be  acquired.  To  do  tliis,  fix  the  eye  on  two 
objects  in  the  desired  line,  such  as  two  trees,  or  bushes,  or  stones, 
or  tufts  of  grass.  Walk  forward,  keeping  the  nearest  of  these 
objects  steadily  covering  the  other.  Before  getting  up  to  the 
nearest  object,  choose  a  new  one  in  line  farther  ahead,  and  then 
proceed  as  before,  and  so  on.  It  is  better  not  to  attempt  to  make 
each  of  the  paces  three  feet,  but  to  take  steps  of  the  natm-al  length, 
and  to  ascertain  the  value  of  each  by  walking  over  a  known  dis- 
tance, and  dividing  it  by  the  number  of  paces  required  to  traverse 
it.  Every  person  should  thus  determine  the  usual  length  of  his 
own  steps,  repeating  the  experiment  suJEciently  often.  The  French 
"  Geographical  Engineers "  accustom  themselves  to  take  regular 


512  SURVEYING  WITHOUT  IXSTRUMEIVTS.      [part  ix. 

steps  of  eiglit-tenths  of  a  metre,  equal  to  two  feet  seven  and  a  half 
inches.  The  English  military  pace  is  two  feet  and  six  inches. 
This  is  regarded  as  a  usual  average.  108  such  paces  per  minute 
give  3.07  English  miles  per  hour.  Quick  pacing  of  120  such  paces 
per  minute  gives  3.41  miles  per  hour.  Slow  paces,  of  three  feet 
each  and  60  per  minute,  give  2.04  miles  per  hour.* 

An  instrument,  called  a  Pedometer,  has  been  contrived,  which 
counte  the  steps  taken  by  one  wearing  it,  without  any  attention  on 
his  part.  It  is  attached  to  the  body,  and  a  cord,  passing  from  it 
to  the  foot,  at  each  step  moves  a  toothed  wheel  one  division,  and 
Bome  intermediate  wheelwork  records  the  whole  number  upon  a  dial. 

(461)  Distances  by  visual  angles*  Prepare  a  scale,  by  marking 
off  on  a  pencil  what  length  of  it,  when  it  is  held  off  at  arm's  length, 
a  man's  height  appears  to  cover  at  different  distances  (previously 
measured  with  accuracy)  of  100,  500,  1000  feet,  &c.  To  apply 
this,  when  a  man  is  seen  at  any  unknown  distance,  hold  up  the 
pencil  at  arm's  length,  making  the  top  of  it  come  in  the  Une  from 
the  eye  to  his  head,  and  placing  the  thumb  nail  in  the  line  from 

Fis.  313. 


the  eye  to  his  feet,  as  in  Fig.  313.  The  pencil  having  been  previ- 
ously graduated  by  the  method  above  explained,  the  portion  of  it 
now  intercepted  between  these  two  lines  will  indicate  the  corre- 
sponding distance. 

If  no  previous  scale  have  been  prepared,  and  the  distance  of  a 
man  be  required,  tajse  a  foot-rule,  or  any  measure  minutely  divided, 
hold  it  off  at  arm's  length  as  before,  and  see  how  much  a  man's 
height  covers.  Then  knowing  the  distance  from  the  eye  to  the 
rule,  a  statement  by  the  Rule  of  Three  (on  the  principle  of  similar 
triangles)  will  give  the  distance  required.  Suppose  a  man's  height, 
of  70  inches,  covers  1  inch  of  the  rule.     He  is  then  70  times  as  far 

•  A  hm-sp   on  a  walk  averages  330  feet  per  minute,  on  a  trot  Q:,Q,  and  on  a  com 
men  gallop  1040.     F  or  longer  times,  the  difference  in  horses  is  more  apparent 


FART  IX.]      J^URFEYIXG  M  ITHOn  L\STRrHENTS.  813 

from  the  eje  as  the  rule ;  and  if  its  distance  be  2  feet,  that  of  the 
man  is  140  feet.  Instead  of  a  man's  height,  that  of  an  ordinary 
house,  of  an  apple-tree,  the  length  of  a  fence-rail,  &c.,  may  be 
be  taken  as  the  standard  of  comparison. 

To  keep  the  arm  immovable,  tie  a  string  of  known  length  to  the 
pencil,  and  hold  between  the  teeth  a  knot  tied  at  the  other  end  of 
the  string. 

(462)  Distances  by  visibility.  The  degree  of  visibility  of  vari- 
ous well-known  objects  will  indicate  approximately  how  far  distant 
they  are.  Thus,  by  ordinary  eyes,  the  windows  of  a  large  house 
can  be  counted  at  a  distance  of  about  13000  feet,  or  2i  miles ; 
men  and  horses  will  be  perceived  as  points  at  about  half  that  dis- 
tance, or  1|  miles  ;  a  horse  can  be  clearly  distinguished  at  about 
4000  feet ;  the  movements  of  men  at  2600  feet,  or  half  a  mile ; 
and  the  head  of  a  man,  occasionally,  at  2300  feet,  and  very  plainly 
at  1300  feet,  or  a  quarter  of  a  mile.  The  Arabs  of  Algeria  define 
a  mile  as  "  the  distance  at  which  you  can  no  longer  distinguish  a 
man  from  a  woman."  These  distances  of  visibility  will  of  course 
vary  somewhat  with  the  state  of  the  atmosphere,  and  still  more  with 
individual  acuteness  of  sight,  but  each  person  should  make  a  corre 
spondmg  scale  for  himself. 

(463)  Distances  by  sound.     Sound  passes  through  the  air  -with 

a  moderate  and  known  velocity  ;  light  passes  almost  instantaneously. 
If,  then,  two  distant  points  be  visible  from  each  other,  and  a  gun 
be  fired  at  night  from  one  of  them,  an  observer  at  the  other,  notmg 
by  a  stop-watch  the  time  at  which  the  flash  is  seen,  and  then  that 
at  which  the  report  is  heard,  can  tell  by  the  intervening  number  of 
seconds  how  far  apart  the  points  are,  knowing  how  far  sound  travels 
in  a  second.  Sound  moves  about  1090  feet  per  second  in  dry  air, 
with  the  temperature  at  the  freezing  point,  32°  Fahrenheit.  For 
higher  or  lowei  temperatures  add  or  subtract  ly  foot  for  each  degree 
of  Fahrenheit  If  a  wind  blows  with  or  against  the  movement  of 
the  sound,  its  velocity  must  be  added  or  subtracted.  If  it  blows 
obliquely,  the  correction  will  evidently  equal  its  velocity  multiplied 
by  the  cosine  of  the  angle  which  the  direction  of  the  wind  makes 


au  SURVEY IXG  IHTMOUT  IIVSTMMEIVTS.       [part  ix. 

mth  the  direction  of  the  sound.*    If  the  gun  be  fired  at  each  end 

of  the  base  in  turn,  and  the  means  of  the  tunes  taken,  the  effect  of 

the  Tvind  will  be  eliminaterl. 

K  a  watch  is  not  at  hand,  suspend  a  pebble  to  a  strmg  (such  aa 

a  thread  drawn  from  a  handkerchief)  and  count  its  vibrations.    If 

it  be  39|  inches  long,  it  will  vibrate  in  one  second ;  if  9|  inches 

long,  in  half  a  second,  &c.     Ef  its  length  is  unknown  at  the  time, 

still  count  its  vibrations  ;   measure  it  subsequently ;   and  then  will 

„      , .       .               T              //lenerth  of  stringV 
the  time  of  its  vibration,  m  seconds,  =  «/ 1 — '^        2 1 . 

(464)  Angles*  Right  angles  are  those  most  frequently  required 
in  this  kind  of  survey,  and  they  can  be  estimated  by  the  eye  with 
much  accuracy.  If  other  angles  are  desu-ed,  they  will  be  deter- 
mined by  measuring  equal  distances  along  the  lines  which  make  the 
angle,  and  then  the  Ime,  or  chord,  joining  the  ends  of  these  distan 
ces,  thus  forming  chain  angles,  explained  in   Art.  (100). 

(465)  Methods  of  operation.  The  "  First  Method"  of  deter 
mining  the  position  of  a  point.  Art.  (5),  is  the  one  most  generality 
applicable.  Some  line,  as  AB  in  Fig.  1,  is  paced,  or  otherwise 
measured,  and  then  the  lines  AS  and  BS  ;  the  point  S  is  thus  de 
termined. 

The  "  Second  Method,"  Art.  (6),  is  also  much  employed,  the 
right  angles  being  obtained  by  eye,  or  by  the  easy  methods  given 
in  Part  II,  Chapter  V,  Arts.  (140),  &c.  It  is  used  for  offsets,  as 
m  Part  II,  Chapter  III,  Arts.  (114),  &c. 

The  "  Third  Method,"  Art.  (7),  may  also  be  used,  the  angles 
being  determined  as  in  Art.  (464). 

The  "  Fourth  Method,"  Art.  (8),  may  also  be  employed,  the 
angles  being  similarly  determined. 

The  "  Fifth  Method,"  Art.  (10),  would  seldom  be  used,  unless 
by  making  an  extempore  plane-table,  and  proceeding  as  directed 
m  the  last  paragraph  of  Art.  (457). 

*  A  gentle,  pleasant  wind  has  a  velocity  of  10  feet  per  second ;  a  bnsk  gale 
20  feet  per  second  ;  a  veiy  brisk  gale  30  feet ;  a  high  wind  50  feet ;  a  veiy  high 
wind  70  feet ;  a  storm  or  tempest  80  feet ;  a  great  storm  100  feet ;  a  hurricana 
120  feet ;  aiid  a  violent  hurricane,  that  tears  up  trees,  &c.,  150  feet  per  second 


PART  IS  ]       SURTETIXG  WITHOri  IXSTRUMEMS. 


315 


The  method  referred  to  in  Art.  (11)  may  also  be  employed. 

WTien  a  sketch  has  made  some  progress,  new  points  may  b^ 
fixed  on  it  by  their  being  in  hne  -with  others  already  determined. 

All  these  methods  of  operation  are  sho-wn  in  the  follo^Ying  figur'' 
AB  is  a  line  paced,  or  otherwise  measured  approximately. 


The  hill  C  is  determined  by  the  fii'st  method.  The  river  on  the 
other  side  of  AB  is  determined  by  offsets  according  to  the 
Second  Method.  The  house  D  is  determined  by  the  Third  Method, 
EBF  being  a  chain  angle.  The  house  G  is  determined  by  the 
Fourth  Method,  chain  angles  being  measured  at  B  and  H,  a  point 
in  AB  prolonged.  The  pond  K  is  determined,  as  in  Art.  (11),  by 
the  intersection  of  the  alinements  CD  and  GH  prolonged.  The 
bend  of  the  river  at  L  is  determined  by  its  distance  from  H  in 
the  Une  of  AH  prolonged.  A  new  base  Hne,  HM,  is  fixed  by  a  chain 
angle  at  H,  and  employed  hke  the  former  one  so  as  to  fix  the  hill 
at  N,  &c.  All  these  methods  may  thus  be  used  collectively  and 
successively.  The  necessary  lines  may  always  be  ranged  with 
rods,  as  directed  in  Art.  (169),  and  very  many  of  the  instrumental 
methods  already  explained,  may  be  practiced  with  extempore  coi>- 
tiivances.  The  use  of  the  Plane-table  is  an  admii-able  prepara- 
tion for  this  style  of  surveying  or  sketching,  which  is  most  fre- 
quently employed  by  Mihtary  Engineers,  though  they  generally 
use  a  prismatic  Compass,  or  pocket  Sextant,  and  a  sketching  case, 
which  may  serve  as  a  Plane-table. 


-t 


^ 


PART  X. 

y  MAPPING. 

CHAPTER  1. 

COPYING  PLATS. 

(466)  The  Flat  of  a  survey  necessarily  has  many  lines  of  construe 
tion  drawn  upon  it,  which  are  not  needed  in  the  finished  map 
These  lines,  and  the  marks  of  instruments,  so  disfigure  the  papei 
that  a  fair  copy  of  the  plat  is  usually  made  before  the  map  is 
finished.  The  various  methods  of  copying  plats,  &c.,  whether  on 
the  same  scale,  or  reduced  or  enlarged,  will  therefore  now  be 
described. 

(467)  Stretching  the  paper.  If  the  map  is  to  be  colored,  the 
paper  must  first  be  wetted  and  stretched,  or  the  application  of  the 
wet  colors  will  cause  its  surface  to  swell  or  blister  and  become  mieven. 
Therefore,  with  a  soft  sponge  and  clean  water  wet  the  back  of  the 
[japer,  working  from  the  centre  outward  in  all  directions.  The 
"  water-mark"  reads  correctly  only  when  looked  at  from  the  front 
Bide,  which  it' thus  distinguishes.  When  the  paper  is  thoroughly 
wet  and  thus  greatly  expanded,  glue  its  edges  to  the  drawing  board, 
for  half  an  inch  in  width,  turning  them  up  against  a  ruler,  passmg 
the  glue  along  them,  and  then  turning  them  down  and  pressing 
them  with  the  ruler.  Some  prefer  gluing  down  opposite  edges  in 
succession,  and  others  adjoining  edges.  The  paper  must  be  mod? 
rately  stretched  smooth  during  the  process.  Hot  glue  is  best. 
Paste  or  gum  may  be  used,  if  the  paper  be  kept  wet  by  a  damp 
clotlij  80  that  the  edges  may  dry  first.     "  Mouth-glue  "  may  be  used 


CHAP,  i]  Copying  Plats.  317 

by  rubbing  it  (moistened  in  the  mouth  or  in  boihng  water)  along  the 
turned  up  edges,  and  then  rubbing  them  dry  bj  an  ivory  folder,  a 
piece  of  dry  paper  being  interposed.  As  this  is  a  slower  process, 
the  middle  of  each  side  should  first  be  fastened  down,  then  the  four 
angles,  and  lastly  the  intermediate  portions.  When  the  paper 
becomes  dry,  the  creases  and  puckerings  will  have  disappeared, 
and  it  will  be  as  smooth  and  tight  as  a  drum-head. 

(468)  Copying  by  tracing.  Fix  a  large  pane  of  clear  glass  in 
a  frame,  so  that  it  can  be  supported  at  any  angle  before  a  window. 
or,  at  night,  in  front  of  a  lamp.  Place  the  plat  to  be  copied  on 
this  glass,  and  the  clean  paper  upon  it.  Connect  them  by  pins, 
&c.  Trace  all  the  desired  hues  of  the  original  with  a  sharp  pencil, 
as  lightly  as  they  can  be  easily  seen.  Take  care  that  the  paper 
does  not  slip.  If  the  plat  is  larger  than  the  glass,  copy  its  parts 
successively,  being  very  careful  to  fix  each  part  in  its  true  relative 
position.  Ink  the  lines  with  India  ink,  making  them  very  fine  and 
pale,  if  the  map  is  to  be  afterwards  colored. 

(469)  Copying  on  tracing  paper.  A  thin  transparent  paper  is 
prepared  expressly  for  the  purpose  of  making  copies  of  maps  and 
drawings,  but  it  is  too  dehcate  for  much  handhng.  It  may  be  pre- 
pared by  soaking  tissue  paper  in  a  mixture  of  turpentine  and 
Canada  balsam  or  balsam  of  fir  (two  parts  of  the  former  to  one  of 
the  latter),  and  drying  very  slowly.  Cold  drawn  hnseed  oil  will 
answer  tolerably,  the  sheets  being  hung  up  for  some  weeks  to  dry. 
Linen  is  also  similarly  pj-epared,  and  sold  under  the  name  of 
"  VeUum  tracing  paper."  It  is  less  transparent  than  the  tracing 
paper,  but  is  very  strong  and  durable.  Both  of  these  are  used 
rather  for  preserving  duplicates  than  for  finished  maps. 

(470)  Copying  by  transfer  paper.  This  is  thm  paper,  one  side 
of  which  is  rubbed  with  blacklead,  &c.,  smoothly  spread  by  cotton. 
It  is  laid  on  the  clean  paper,  the  blackened  side  downward,  and 
the  plat  is  placed  upon  it.  All  the  lines  of  the  plat  are  then  gone 
over  with  moderate  pressure  by  a  blunt  point,  such  as  the  eye-end 
jf  a  small  needle.     A  faint  tracino:  of  these  lines  will  then  be  found 


318  MAPPING.  [part  x. 

on  the  clean  paper,  and  can  be  inked  at  leisure.  If  the  originai 
cannot  be  thus  treated,  it  may  first  be  copied  on  tracing  paper, 
and  this  copy  be  thus  transferred.  If  the  transfer  paper  be  pre- 
pared by  rubbing  it  with  lampblack  ground  up  with  hard  soap,  its 
lines  will  be  ineffaceable.     It  is  then  called  "  Camp-paper." 

(471)  Copying  by  punctures.  Fix  the  clean  paper  on  a  draw- 
ing board  and  the  plat  over  it.  Prepare  a  fine  needle  with  a  seal- 
ing-wax head.  Hold  it  very  truly  perpendicular  to  the  board,  and 
prick  through  every  angle  of  the  plat,  and  every  corner  and  inter- 
section of  its  other  lines,  such  as  houses,  fences,  &c.,  or  at  least 
the  two  ends  of  every  line.  For  circles,  the  centre  and  one  point 
of  the  circumference  are  sufficient.  For  irregular  curves,  such  as 
rivers,  &c.,  enough  points  must  be  pricked  to  indicate  all  their 
sinuosities.  Work  with  system,  finishing  up  one  strip  at  a  time, 
so  as  not  to  omit  any  necessary  points  nor  to  prick  through  any 
twice,  though  the  latter  is  safer.  When  completed,  remove  the 
plat.  Tlie  copy  will  present  a  wilderness  of  fine  pomts.  Select 
those  which  determine  the  leading  lines,  and  then  the  rest  will  be 
easily  recognized.  A  beginner  should  first  pencil  the  lines  lightly, 
and  then  ink  them.  An  experienced  draftsman  will  omit  the  pen- 
cilling. Two  or  three  copies  may  be  thus  pricked  through  at  once. 
The  holes  in  the  original  plat  may  be  made  nearly  invisible  by 
rubbing  them  on  the  back  of  the  sheet  with  a  paper-folder,  or  the 
thumb  nail. 

(472)  Copying  by  intersections.  Draw  a  line  on  the  clean  paper 
equal  in  length  to  some  important  Une  of  the  original.  Two  start- 
ing points  are  thus  obtained.  Take  in  the  dividers  the  distance 
from  one  end  of  the  line  on  the  original  to  a  third  pomt.  From 
the  corresponding  end  on  the  copy,  describe  an.  arc  with  this  dis- 
tance for  radius  and  about  Avhere  the  point  will  come.  Take  the 
distance  on  the  original  from  the  other  end  of  the  line  to  the  point, 
and  describe  a  corresponding  arc  on  the  copy  to  intersect  the 
former  arc  in  a  point  which  will  be  that  desired.  The  prmciple 
of  the  operation  is  that  of  our  "  First  Method,"  Art.  (5).  Two 
Dairs  of  dividers  may  be  used  as  explained  in  Art.  (90).     "  Tri- 


CHAP.  I.]  Copjing  Plats.  319 

angular  compasses,"  having  three  legs,  are  used  by  fixing  two  of 
their  legs  on  the  two  given  points  of  the  original,  and  the  third  leg 
on  the  point  to  be  copied,  and  then  transferring  them  to  the  copy. 
All  the  points  of  the  origuial  can  thus  be  accurately  reproduced. 
The  operation  is  however  very  slow.  Only  the  chief  points  of  a 
plat  may  be  thus  transferred,  and  the  details  filled  in  by  the  fol- 
lowing method. 

(473)  Copying  by  squares.  On  the  original  plat  draw  a  series 
of  parallel  and  equidistant  lines.  The  T  square  does  this  most 
readily.  Draw  a  similar  series  at  right  angles  to  these.  The  plat 
will  then  be  covered  with  squares,  as  in  Fig.  38,  page  48.  On 
the  clean  paper  draw  a  similar  series  of  squares.  The  important 
points  may  now  be  fixed  as  in  the  last  article,  and  the  rest  copied 
by  eye,  all  the  points  in  each  square  of  the  original  being  properly 
placed  in  the  corresponding  square  of  the  copy,  noticing  whether 
they  are  near  the  top  or  bottom  of  each  square,  on  its  right  or  left 
Bide,  &c.  This  method  is  rapid,  and  in  skilful  hands  quite  accu- 
rate. 

Instead  of  drawing  lines  on  the  original,  a  sheet  of  transparent 
paper  containing  them  may  be  placed  over  it ;  or  an  open  frame 
with  threads  stretched  across  it  at  equal  distances  and  at  right 
angles. 

This  method  supplies  a  transition  to  the  Reduction  and  Enlarge- 
ment of  plats  in  any  desired  ratio  ;  under  which  headCopym^by 
the  Pantagraph  and  Camera  Lucida  will  be  noticed. 

(474)  Redncing  by  squares.  Begin,  as  in  the  preceding  article, 
by  drawing  squares  on  the  original,  or  placing  them  over  it.  Then 
on  the  clean  paper  draw  a  similar  set  of  squares,  but  with  their 
sides  one-half,  one  third,  &c.,  (according  to  the  desired  reduction), 
of  those  of  the  original  plat.  Then  proceed  as  before  to  copy  into 
each  small  square  all  the  points  and  lines  found  in  the  large  square 
of  the  plat  in  their  true  positions  relative  to  the  sides  and  corners 
of  the  square,  observing  to  reduce  each  distance,  by  eye  or  as 
directed  in  the  following  article,  in  the  given  ratio. 


820 


MAPPING. 


[part  X. 


(475)  Reducing  hy  proportional  scales.  Many  graphical  me- 
thods of  finding  the  proportionate  length  on  the  copy,  of  any  line  of 
the  original,  may  be  used.  The  "Angle  of  reduction"  is  con 
structed   thus.     Draw   any   Hne  \c 

AB.     With  it  for  radius  and  A  fig.  315. 

for  centre,  describe  an  indefinite 
arc.  With  B  for  centre  and  a 
radius  equal  to  one-half,  one-third, 
&c.,  of  AB  according  to  the  de- 
sired reduction  describe  another 
arc  intersecting  the  former  arc 
centre  describe  a  series  of  arcs 


Fig.  316. 


A  B  B 

in  C.  Join  AC.  From  A  aa 
Now  to  reduce  any  distance, 
take  it  in  the  dividers,  and  set  it  off  from  A  on  AB,  as  to  D.  Then 
the  distance  from  D  to  E,  the  other  end  of  the  arc  passing  through 
J),  will  be  the  proportionate  length  to  be  set  off  on  the  copy,  in  the 
manner  directed  in  Art.  (472). 

The  Sector,  or  "  Compass  of  proportion,"  described  in  Art.  (52), 
presents  such  an  "Angle  of  reduction,"  always  ready  to  be  used 
in  this  manner. 

The  "  Angle  of  reduction"  may  be  simplified 
thus.  Draw  a  line,  AB,  parallel  to  one  side 
of  the  drawing  board,  and  another,  BC,  at  right 
angles  to  it,  and  one-half,  &c.,  of  it,  as  desired. 
Join  AC.  Then  let  AD  be  the  distance  re- 
quired to  be  reduced.  Apply  a  T  square  so 
as  to  pass  through  D.  It  will  meet  AC  in 
some  point  E,  and  DE  will  be  the  reduced 
length  required. 

Another  arrangement  for  the  same  object  is  shown  in  Fig.  317. 
Draw  two  lines,  AB,  AC,  at  any  angle,  and  de- 
scribe a  series  of  arcs  from  their  intersection.  A, 
as  in  the  figure.  Suppose  the  reduced  scale  is  to  ^ 
be  half  the  original  scale.  Divide  the  outermost 
arc  into  three  equal  parts,  and  draw  a  line  from 
A  to  one  of  the  points  of  division,  as  D.  Then 
each  arc  will  be  divided  into  parts,  one  of  which 
is  twice  the  other.  Take  any  distance  on  the  ori- 
ginal scale,  and  find  by  trial  which  of  the  arcs  on 


Fig.  317. 


CHAP,  i]  Copying  Plats.  3?1 

the  right  hand  side  of  the  figure  it  corresponds  to.     The  other  part 
of  that  arc  -will  be  half  of  it,  as  desired. 

"  Proportional  compasses,"  being  properly  set,  reduce  lines  in 
any  desired  ratio.  A  simple  form  of  them,  known  as  "  Wholes 
and  halves,"  is  often  useful.  It  consists  of  two  slender  bars,  pointed 
at  each  end,  and  united  by  a  pivot  which  is  twice  as  far  from 
one  pair  of  the  points  as  from  the  other  pair.  The  long  ends  bemg 
set  to  any  distance,  the  short  ends  will  give  precisely  half  that  dis- 
tance. 

(476)  Reducing  by  a  pantagraph.  This  instrument  consists  of 
two  long  and  two  short  rulers,  connected  so  as  to  form  a  parallelo- 
gram, and  capable  of  being  so  adjusted  that  when  a  tracing  point 
attached  to  it  is  moved  over  the  lines  of  a  map,  &c.,  a  pencil 
attached  to  another  part  of  it  will  mark  on  paper  a  precise  copy, 
reduced  on  any  scale  desired.  It  is  made  in  various  forms.  It  is 
troublesome  to  use,  though  rapid  in  its  work. 

(477)  Reducing  by  a  camera  lucida.  This  is  used  in  the  Coast 
Survey  Office.  It  cannot  reduce  smaller  than  one-fourth,  without 
losing  distinctness,  and  is  very  trying  to  the  eyes.  Squares  drawn 
on  the  original  are  brought  to  apparently  coincide  with  squares  on 
the  reduction,  and  the  details  are  then  filled  in  with  the  pencil,  as 
Been  through  the  prism  of  the  instrument. 

(478)  Enlarging  plats.  Plats  may  be  enlarged  by  the  princi- 
pal methods  which  have  been  given  for  reducing  them,  but  this 
should  be  done  as  seldom  as  possible,  suice  every  inaccuracy  in  the 
original  becomes  magnified  in  the  copy.  It  is  better  to  make  • 
new  plat  from  the  origmal  data. 


21 


'6^1  MAPPING.  [PAKi  X 


CHAPTER  II. 


tOMENTIONAL  SIGNS. 

(479)  Various  conventional  signs  or  marks  have  been  adopted, 
more  or  less  generally,  to  represent  on  maps  the  inequalities  of  the 
surface  of  the  ground,  its  different  kinds  of  culture  or  natural  pro- 
ducts, and  the  objects  upon  it,  so  as  not  to  encumber  and  disfigure 
it  with  much  wi'iting  or  many  descriptive  legends.  This  is  the 
purpose  of  what  is  called  Topographieal  Mapping. 

(480)  The  relief  of  ground.  The  inequalities  of  the  surface 
of  the  earth,  its  elevations  and  depressions,  its  hills  and  hollows, 
constitute  its  "  Relief."  The  representation  of  this  is  sometimes 
called  "  HiU  drawing."  Its  difficulty  arises  from  our  being  accus- 
tomed to  see  hills  sideways,  or  "  in  elevation,"  while  they  must 
be  represented  as  they  would  be  seen  from  above,  or  "  in  plan." 
Various  modes  of  thus  drawmg  them  are  used ;  their  positions  being 
laid  down  in  pencil  as  previously  sketched  by  eye  or  measured. 

If  light  be  supposed  to  fall  vertically,  the  slopes  of  the  ground  will 
receive  less  light  in  proportion  to  their  steepness.  The  rehef  of 
ground  will  be  indicated  on  this  principle  by  making  the  steep 
slopes  very  dark,  the  gentler  inclinations  less  so,  and  leavuig  the 
level  surfaces  white.  The  shades  may  be  produced  by  tints  of 
India  ink  apphed  with  a  brush,  their  edges,  at  the  top  and  'ottom 
of  a  hill  or  ridge,  being  softened  off  with  a  clean  brush. 

If  hght  be  supposed  to  fall  obliquely,  the  slopes  facing  it  will  be 
light,  and  those  turned  from  it  dark.  This  mode  is  effective,  but 
not  precise.  In  it  the  hght  is  usually  supposed  to  come  from  the 
upper  left  hand  corner  of  the  map. 

Horizontal  contour  lines  are  however  the  best  convention  for 
this  purpose.  Imagine  a  hill  to  be  sUced  off  by  a  number  of  equi- 
distant horizontal  planes,  and  their  intersections  with  it  to  be  drawn 
as  they  would  be  seen  from  above,  or  horizontally  projected  on  the 


CUAP  II.] 


CoKvcntioual  Signs. 


323 


map.  These  are  "  Contour  lines."  Tliej  are  the  same  lines  as 
would  be  formed  by  water  surrounding  the  hill,  and  rising  one  foot 
at  a  time  (or  any  other  height)  till  it  reached  the  top  of  the  hill. 
The  edge  of  the  water,  or  its  shore,  at  each  successive  rise,  would 
be  one  of  these  horizontal  contour  lines.  It  is  plain  that  their 
nearness  or  distance  on  the  map  would  indicate  the  steepness  or 
gentleness  of  the  slopes.      A  right  cone  would  thus  be  repre- 

Fii-'.  318.  Fig.  319.  Fig.  320. 


sented  by  a  series  of  concentric  circles,  as  in  Fig.  318  ;  an  oblique 
cone  by  circles  not  concentric,  but  nearer  to  each  other  on  the  steep 
side  than  on  the  other,  as  in  Fig.  319  ;  and  a  half-egg,  somewhat 
as  in  Fig.  320. 

Vertical  sections,  perpendicular  to  these  contour  lines,  are 
usually  combined  with  them.  They  are  the  "  Lines  of  greatest 
?lope,"  and  may  be  supposed  to  represent  water  running  down  the 
sides  of  the  hill.  They  are  also  made  thicker  and  nearer  together 
on  the  steeper  slopes,  to  produce  the  effect  required  by  the  conven- 
tion of  vertical  Hght  ^ig-  32 1. 
already  referred  to.  ..,«\\\\\\\M 
The  marginal  fiorure 
shews  an  elongated 
half-egg,  or  oval  hill,  ^^ 


thus  represented. 

The  spaces  between 
the  rows  of  vertical 
"  Hatchings"  indicate 
the  contour  lines,  which  are  not  actually  drawn.  The  beauty  of 
the  graphical  execution  of  this  work  depends  on  the  uniformity  of 
the  strokes  representing  uniform  slopes,  on  their  perfectly  regular 
gradation  in  thickness  and  nearness  for  varying  slopes,  and  on 
their  being  made  precisely  at  right  angles  to  the  contour  linea 
between  which  they  are  situated. 


B24  MAPPL\G.  [partx 

The  methods  of  determining  the  contour  lines  are  applications 
of  Levelling,  and  will  therefore  be  postponed,  together  with  the 
farther  details  of  "  Hill-drawing,"  to  the  volume  trea'-hig  of  that 
subject,  which  is  announced  in  the  Preface. 

(481)  Signs  for  natural  surface.  Sand  is  represented  by  fine 
dots  made  with  the  point  of  the  pen ;  gravel  by  coarser  dots. 
Bocks  are  drawn  in  their  proper  places  in  irregular  angular  forms, 
imitating  their  true  appearance  as  seen  from  above.  The  nature 
of  the  rocks,  or  the  G-eology  of  the  country,  may  be  shown  by  apply- 
ing the  proper  colors,  as  agreed  on  by  geologists,  to  the  back  of 
the  map,  so  that  they  may  be  seen  bj'  holding  it  up  against  the 
light,  while  they  will  thus  not  confuse  the  usual  details. 

(482)  Signs  for  vegetation.      Woods  are  represented  by  scol- 
loped circles,  irregularly  disposed,  Fig.  322. 
imitating  trees  seen  "  in  plan,"  and    <^-  ^'^^h-  ^  <:^'  ^'  ^ 
closer  or  farther  apart  according  to   A^^M  ^^Q}M  ^%^:^  tv# 
the  thickness  of  the  forest.     It  is   '^a^.^&^^-^P'W.. 


usual  to  shade  their  lower  and  right  ^^^^t^^^^^^^ 
hand  sides  and  to  represent  their  wX^  '^'^Q^^^*^'^  w-^ 
shadows,  as  in  the  figure,  though,  in  strictness,  this  is  inconsistent 
with  the  hypothesis  of  vertical  light,  adopted  for  "  hill-drawing." 
For  pine  and  similar  forests,  the  signs  may  have  a  star-like  form, 
as  on  the  right  hand  side  of  the  figure.  Trees  are  sometimes 
drawn  "  in  elevation,"  or  sideways,  as  usually  seen.  This  makes 
them  more  easily  recognized,  but  is  in  utter  violation  of  the  princi- 
ples of  mapping  in  horizontal  projection,  though  it  may  be  defended 
as  a  pure  convention.  Orchards  are  represented  by  trees  arrang- 
ed in  rows.  Bushes  may  be  drawn  like  trees,  but  smaller. 
Grass-land  is  drawn  with  irregularly  Fig.  323. 


<,\ttK. 


scattered  groups  of  short  lines,  as  in  the    -""^^i;;'--"'-^^-*"^^^'*''  yVV 


figure,  the  lines  being  arrano;ed   in  odd    "'"'^^»~ I«»-"^^l^'^'X,iTi.    ^,  ,!w 
numbers,  and  so  that  the  top  of  each  group  is     :iit'""-i'^'-'^C.ir^j«l    v  v\f 
convex  and  its  bottom  horizontal  or  parallel    ^«"'.''''!i3^""-  '""'^-*^;  -^  mmv 
to  the  base  of  the  drawing.     Meadows  are     *^  *'  ^-  ■^'"^  ""^  ^ 
sometimes  represented  by  pairs  of  diverging  lines,  (as  on  the  right 


j  m!  lil  liiii 
iiilliliillilil! 


CHAP.  II  ]  Couveiitioual  Signs.  325 

of  the  figure),  which  may  be  regarded  as  tall  blades  of  grass. 
Uncultivated  land  is  indicated  by  appropriately  intermingling  the 
signs  for  grass  land,  bushes,  sand  and  rocks.  Cultivated  land  is 
shown  by  parallel  rows  of  broken  and  dotted  i''g-  ^~i- 

lines,  as  in  the  figure,  representing  furrows. 
Crops  are  so  temporary  that  signs  for  them  are 
unnecessary,  though  often  used.  They  are  usu- 
ally imitative,  as  for  cotton,  sugar,  tobacco,  rice,  ^^^Zr^^sr~-~~ 
vines,  hops,  &c.  Gardens  are  drawn  with  cir-  E-I-Z~~z:i:'E:'iJ2rl^ 
cular  and  other  beds  and  walks. 

(483)  Signs  for  water.  The  Sea-coast  is  xepresented  by  draw- 
ing a  line  parallel  to  the  shore,  following  all  its  windings  and  inden- 
tations, and  as  close  to  it  as  possible,  then  another  parallel  line  a 
little  more  distant,  then  a  third  still  more  distant,  and  so  on. 
Examples  are  seen  in  figures  287,  &c.  If  these  Imes  are  drawn 
from  the  low  tide  mark,  a  similar  set  may  be  drawn  between  that 
and  the  high  tide  mark,  and  dots,  for  sand,  be  made  over  the 
included  space.  Rivers  have  each  shore  treated  like  the  sea 
shore,  as  in  the  figures  of  Part  VII.*  Brooks  would  be  shown  by 
only  two  lines,  or  one,  according  to  their  magnitude.     Ponds  may 

be  drawn  hke  sea  shores,  or  represented  by Fig,  sg.i. 

parallel  horizontal  lines  ruled  across  them. 
Marshes  and  Swamps  are  represented  by  an 
irregular  intermingling  of  the  preceding 
sign  with  that  for  grass  and  bushes,  as  in  the 
figure.  

(484)  Colored  Topography.  The  conventional  signs  which  have 
been  described,  as  made  with  the  pen,  require  much  time  and 
labor.  Colors  are  generally  used  by  the  French  as  substitutes  for 
them,  and  combine  the  advantages  of  great  rapidity  and  efiective- 
ness.  Only  three  colors  (besides  India  ink)  are  required ;  viz. 
Gamboge  (yellow),  Indigc  (blue),  and  Lahe  (pink).  Sepia, 
Burnt  Sienna,  Yellow  ochre.  Red  lead,  and  Vermillion,  are  also 
sometimes  used.     The  last  three  are  difficult  to  work  with.     Tc 

*  Those  ill  Part  II,  Cliupter  V,  have  the  lines  loo  close  togelhet  in  the  middle. 


326  MAPPIXG.  [PABT  X 

use  these  paints,  moisten  the  end  of  a  cake  and  rub  it  up  with  a 
drop  of  water,  afterwards  diluting  this  to  the  proper  tint,  which 
should  always  be  light  and  delicate.  To  cover  any  surface  with 
a  uniform  flat  tint,  use  a  large  camel's  hair  or  sable  brush,  keep  it 
always  moderately  full,  incline  the  board  towards  you,  previously 
moisten  the  paper  with  clean  water  if  the  outline  is  very  irregular, 
begin  at  the  top  of  the  surface,  apply  a  tint  across  the  upper  part. 
and  continue  it  downwards,  never  letting  the  edge  dry.  This  last 
is  the  secret  of  a  smooth  tint.  It  requires  rapidity  in  returning  to 
the  beginning  of  a  tint  to  continue  it,  and  dexterity  in  follo-^ing  the 
outline.  Marbling,  or  variegation,  is  produced  by  having  a  brush 
at  each  end  of  a  stick,  one  for  each  color,  and  applying  first  one, 
and  then  the  other  beside  it  before  it  dries,  so  that  they  may  blend 
but  not  mix,  and  produce  an  irregularly  clouded  appearance. 
Scratched  parts  of  the  paper  may  be  painted  over  by  first  applying 
strong  alum  water  to  the  place. 

The  conventions  for  colored  Topography,  adopted  by  the  French 
Military  Engineers,  are  as  follows.  Woods,  yellow  ;  using  gam- 
boge and  a  very  httle  indigo.  Grass-land,  green;  made  of 
gamboge  and  indigo.  Cultivated  land,  hrown;  lake,  gamboge, 
and  a  little  India  ink.  "  Burnt  Sienna"  will  answer.  Adjoining 
fields  should  be  shghtly  varied  in  tuit.  Sometimes  furrows  are 
indicated  by  strips  of  various  colors.  Gardens  are,  represented 
by  small  rectangular  patches  of  brighter  green  and  hrown.  Un- 
cultivated land,  marbled  green  and  light  hrown.  Brush, 
brambles,  &c.,  marbled  green  and  yellow.  Heath,  furze,  &c., 
marbled  green  and  pink.  Vineyards,  purple  ;  lake  and  indigo. 
Sands,  a  Ught  hroivn ;  gamboge  and  lake.  "  Yellow  ochre"  will 
io.  Lakes  and  rivers,  light  blue,  with  a  darker  tint  on  their 
upper  and  left  hand  sides.  Seas,  dark  hlue,  with  a  little  yellow 
added.  Marshes,  the  blue  of  water,  with  spots  of  grass  green,  the 
touches  all  lying  horizontally.  Roads,  hroivn  ;  between  the  tints 
for  sand  and  cultivated  ground,  with  more  India  ink.  Hills, 
ff^emish  brown  ;  gamboge,  indigo,  lake  and  India  ink,  instead  of 
the  pure  India  ink,  directed  in  Art.  (480).  Woods  may  be 
finished  up  by  di-a^ying  the  trees  as  in  Art.  (482)  and  coloring 
them  green,  with  touches  of  gamboge  towards  the  light  (the  uppei 
and  left  hand  side)  and  of  indigo  on  the  opposite  side. 


CHAP.  II.] 


Conventional  Sisnsi 


327 


(485)  Signs  for  detached  objects.  Too  great  a  number  of  these 
will  cause  confusion.  A  few  leading  ones  will  be  given,  the  mean- 
ings of  which  are  apparent. 


Court  house, 

Figs. 
gt|    326. 

Wind  mill. 

Figs. 
@X334. 

Fost  office, 

Steam  mill, 

^335. 

Tavern, 

. 

^     328. 

Furnace, 

1    336 

Blacksmith^s 

shop. 

&    329. 

Woollen  factory, 

#  337 

G-uide  hoard. 

t      330. 

Cotton  factory, 

^  338 

Quarry, 

X     331. 

Glass  works. 

M  339 

Grist  mill. 

O      332. 

Church, 

^    34C 

Saw  mill,  >^    333.  Grave  yard,        --JX.34I 

An  ordinary  house  is  drawn  in  its  true  position  and  size,  and  /l»e 
ridge  of  its  roof  shown  if  the  scale  of  the  map  is  large  eno'  .gh. 
On  a  very  small  scale,  a  small  shaded  rectangle  represents  it.  If 
colors  are  used,  buildings  of  masonry  are  tinted  a  deep  crimson, 
(with  lake),  and  those  of  wood  with  India  ink.  Their  lower  and 
right  hand  sides  are  drawn  with  heavier  lines.  Fences  of  stone  or 
wood,  and  hedges,  may  be  drawn  in  imitation  of  the  reahties ;  and, 
if  desired,  colored  appropriately. 

Mines  may  be  represented  by  the  signs  of  the  planets  which 
were  anciently  associated  with  the  various  metals.  The  signs  here 
given  represent  respectively. 

Gold,     Silver,     Iron,     Copper,     Tin,     Lead,     Quicksilver. 

A  large  black  circle,  #  ,  may  be  used  for  Coal. 

Boundary  lines,  of  private  properties,  of  townships,  of  counties, 
and  of  states,  may  be  indicated  by  lines  f-^rmed  of  various  combi- 
nations of  short  lines,  dots  and  crosses,  as  below.* 


+  +  +  +   +   +  +  +  +  +  +  +  +   +  +  +  +  + 

•  Very  minute  directions  for  the  execution  of  tiie  details  described  in  this  chap 
tor,  are  given  in  Lieut.  R.  S.  Smith's  "  Topographical  Drawing."     Wiley    N.  Y. 


828  MAPPING.  fpABix 


CHAPTER  in. 


FIIVISHIIVG  THE  MAP. 

(486)  Orientation.  The  map  is  usually  so  drawn  that  the  top 
of  the  paper  may  represent  the  North.  A  Meridian  Une  should 
also  be  drawn,  both  True  and  Magnetic,  as  in  Fig.  199,  page  189. 
The  number  of  degrees  and  minutes  in  the  Variation,  if  known, 
should  also  be  placed  between  the  two  North  points.  Sometimes 
a  compass-star  is  drawn  and  made  very  ornamental. 

(487)  Lettering.  The  style  in  which  this  is  done  very  much 
affects  the  general  appearance  of  the  map.  The  young  surveyor 
should  give  it  much  attention  and  careful  practice.  It  must  all  be 
in  imitation  of  the  best  printed  models.  No  writing,  however 
beautiful,  is  admissible.  The  usual  letters  are  the  ordinary 
ROMAN  CAPITALS,  SmaU  Roman,  ITALIC  CAPITALS, 
Small  Italic,  and  GOTHIC  OR  EGYPTIAN,  This  last, 
when  well  done,  is  very  effective.  For  the  Titles  of  maps,  various 
fancy  letters  may  be  used.  For  very  large  letters,  those  formed 
only  of  the  shades  of  the  letters  regarded  as  blocks  (the  body  being 
rubbed  out  after  being  pencilled  as  a  guide  to  the  placing  of  the 
shades)  are  most  easily  made  to  look  well.  The  simplest  lettering 
is  generally  the  best.  The  sizes  of  the  names  of  places,  &c.,  should 
be  proportional  to  their  importance.  Elaborate  tables  for  various 
scales  have  been  published.  It  is  better  to  make  the  letters  too 
small  than  too  large.  They  should  not  be  crowded.  Pencil  lines 
should  always  be  ruled  as  guides.  The  lettering  should  be  in  lines 
parallel  to  the  bottom  of  the  map,  except  the  names  of  rivers,  roads, 
&c.,  whose  general  course  should  be  followed. 

(488)  Borders.  The  Border  may  be  a  single  heavy  line, 
enclosing  the  map  in  a  rectangle,  or  such  a  line  may  be  relieved 
by  a  finer  line  drawn  parallel  and  near  to  it.  Time  should  not  be 
wasted  in  ornamenting  the  border.     The  simplest  is  the  best. 


CBAP  III]  Finisliing  the  Map,  32S 

(489)  Joining  paper.  If  the  map  is  larger  than  the  sheets  of 
paper  at  hand,  they  should  be  joined  vi'ith  a  feather-edge,  by  pro* 
ceeding  thus.  Cut,  with  a  knife  guided  by  a  ruler,  about  one- 
third  through  the  thickness  of  the  paper,  and  tear  off  on  the  under 
side,  a  strip  of  the  remaining  thickness,  so  as  to  leave  a  thin  sharp 
edge.  Treat  the  other  sheet  in  the  same  way  on  the  other  side  of 
it.  When  these  two  feather  edges  are  then  put  together,  (with 
paste,  glue  or  varnish),  they  will  make  a  neat  and  strong  joint. 
The  sheet  which  rests  upon  the  other  must  be  on  the  right  hand 
side,  if  the  sheets  are  joined  lengthways,  or  below  if  they  are  joined 
in  that  direction,  so  that  the  thickness  of  the  edge  may  not  cast  a 
shadow,  when  properly  placed  as  to  the  hght.  The  sheets  must 
be  joined  before  lines  are  drawn  across  them,  or  the  lines  will 
become  distorted.  ')rawing  paper  is  now  made  in  rolls  of  great 
length,  so  as  to  render  this  operation  unnecessary. 

(490)  Mounting  maps,  A  map  is  sometimes  required  to  be 
mounted,  i.  e.  backed  with  canvas  or  muslin.  To  do  this,  wet  the 
muslui  and  stretch  it  strongly  on  a  board  by  tacks  driven  very 
near  together.  Cover  it  with  strong  paste,  beating  this  in  with  a 
brush  to  fill  up  the  pores  of  the  muslin.  Then  spread  paste  over 
the  back  of  the  paper,  and  when  it  has  soaked  into  it,  apply  it  to 
the  muslin,  inclining  the  board,  and  pasting  first  a  strip,  about  two 
inches  wide,  along  the  upper  side  of  the  paper,  pressing  it  down 
with  clean  linen  in  order  to  drive  out  all  air  bubbles.  Press  down 
another  strip  in  like  manner,  and  so  proceed  till  all  is  pasted.  Let 
it  dry  very  gradually  and  thoroughly  before  cutting  the  muslin 
from  the  board. 

Maps  may  be  varnished  with  picture  varnish ;  or  by  applying 
four  or  five  coats  of  isinglass  size,  letting  each  dry  well  before 
applying  the  next,  and  giving  a  full  flowing  coat  of  Canada  balsam 
diluted  with  the  best  oil  of  turpentine. 


PAET  XI. 

LAYING    OUT,    PARTING   OFF,   AND 
DIVIDING    UP    LAND.* 

CHAPTER  I. 

LAYING  OIT  LAIVD. 

(491)  lis  nature.  This  operation  is  precisely  the  reverse  ot 
those  of  Surveying  properly  so  called.  The  latter  measures  certain 
lines  as  they  are ;  the  former  marks  them  out  in  the  ground  where 
they  are  required  to  be,  in  order  to  satisfy  certain  conditions. 
The  same  instruments,  however,  are  used  as  in  Surveying. 

Perpendiculars  and  parallels  are  the  lines  most  often  employed. 
The  Perpendiculars  may  be  set  out  either  with  the  chain  alone, 
Arts.  (140)  to  (159)  ;  still  more  easily  with  the  Cross-staff,  Art 
(104),  or  the  Optical-square,  Art.  (107)  ;  and  most  precisely  with 
a  Transit  or  Theodolite,  Arts.  (402)  to  (406).  Parallels  may 
also  be  set  out  with  the  chain  alone.  Arts.  (160)  to  (166)  ;  or 
with  Transit,  &c..  Arts.  (407)  and  (408).  The  ranging  out  of 
lines  by  rods  is  described  in  Arts.  (169)  and  (178),  and  with  an 
Angular  instrument,  in  Arts.  (376),  (409)  and  (415). 

(492)  To  lay  out  squares.  Reduce  the  desired  content  to 
square  chains,  and  extract  its  square  root.  This  wUl  be  the  length 
of  the  required  side,  which  is  to  be  set  out  by  one  of  the  methods 
indicated  in  the  preceding  article. 

An  Acre,  laid  out  in  the  form  of  a  square,  is  frequently  desired 
by  farmers.     Its  side  must  be  made  816^  links  of  a  Gunter's 

*  The  Demonslrations  of  the  Problems  in  this  part,  when  required  will  b« 
fouud  in  Ajipenilix  B. 


CHAP.  I.]  Layiug  out  Laud.  831 

chain  ;  or  208i'yL  feet ;  or  G9j^q  yards.     It  is  often  taken  at 
70  paces. 

The  number  of  plants,  hills  of  corn,  loads  of  manure,  &c.,  which 
an  acre  will  contain  at  any  uniform  distance  apart,  can  be  at  once 
found  by  dividing  209  by  this  distance  in  feet,  and  multiplying 
the  quotient  by  itself ;  or  by  dividing  43560  by  the  square  of  the 
distance  in  feet.  Thus,  at  3  feet  apart,  an  acre  would  contain 
4840  plants,  &c. ;  at  10  feet  apart,  436  ;  at  a  rod  apart,  160  ; 
and  so  on.  If  the  distances  apart  be  unequal,  divide  43560  by 
the  product  of  these  distances  in  feet ;  thus,  if  the  plants  were  in 
rows  6  feet  apart,  and  the  plants  in  the  rows  were  3  feet  apart, 
2420  of  them  would  grow  on  one  acre. 

(493)  To  lay  out  rectaugles.  The  content  and  length  being 
given^  both  as  measured  by  the  same  unit,  divide  the  former  by 
the  latter,  and  the  quotient  will  be  the  required  breadth.  Thus,  1 
acre  or  10  square  chains,  if  5  chains  long,  must  be  2  chauas  wide. 

The  content  being  given  and  the  length  to  be  a  certain  number 
of  times  the  breadth.  Divide  the  content  in  square  chains,  &c.,  by 
the  ratio  of  the  length  to  the  breadth,  and  the  square  root  of  the 
quotient  -will  be  the  shorter  side  desired,  -whence  the  longer  side 
is  also  known.  Thus,  let  it  be  required  to  lay  out  30  acres  in  the 
form  of  a  rectangle  3  times  as  long  as  broad.  30  acres  =  300 
square  chains.  The  desired  rectangle  will  contain  3  squares,  each 
of  100  sq.  chs.,  having  sides  of  10  chs.  The  rectangle  will  there- 
fore be  10  chs.  wide  and  30  long. 

An  Acre  laid  out  in  a  rectangle  twice  as  long  as  broad,  -vsill  be 
B^Hnks  b^  ^hnks,  n?arly ;  or  147^  feet  by  295  feet ;  or  49^ 
yards  by  98|  yardsr  50  paces  by  100  is  often  used  as  an  ap- 
proximation, easy  to  be  remembered. 

The  content  being  given,  and  the  difference  between  the  length 
and  breadth.  Let  c  represent  this  content,  and  d  this  difference. 
Then  the  longer  side  =  i  (i  +  |  VC*^^  +  4  :?). 

Example.  Let  the  content  be  6.4  acres,  and  the  difference 
12  chains.  Then  the  sides  of  the  rectangle  will  be  respeciiv^'lj 
16  chains  and  4  chains. 


232  LAYI\«  OUT  KM)  DIVIDIXG  UP  LAND.       [part  xi 

The  content  being  given,  and  the  sum  of  the  length  and  breadth. 
Let  c  represent  this  content,  and  s  tliis  sum.  Then  the  longei 
8icle=|s+  1  ^(s2  — ic). 

Example.  Let  the  content  be  6.4  acres,  and  the  sum  20  chains. 
The  above  formula  gives  the  sides  of  the  rectangle  16  chains  and  4 
chains  as  before. 

(494)  To  lay  out  triangles.  The  content  and  the  base  being 
given,  divide  the  former  by  half  the  latter  to  get  the  height.  At 
any  point  of  the  base  erect  a  perpendicular  of  the  length  thua 
obtained,  and  it  will  be  the  vertex  of  the  required  triangle. 

The  content  being  given  and  the  base  having  to  be  rn  times  the 
height,  the  height  will  equal  the  square  root  of  the  quotient 
obtained  by  dividing  twice  the  given  area  by  m. 

The  content  being  given  and  the  triangle  to  be  equilateral,  take 
the  square  root  of  the  content  and  multiply  it  by  1.520.  The  pro- 
duct will  be  the  length  of  the  side  required.  This  rule  makes  the 
sides  of  an  equilateral  triangle  containing  one  acre  to  be  480i  links. 
A  quarter  of  an  acre  laid  out  in  the  same  form  would  have  each 
side  240  hnks  long.  An  equilateral  triangle  is  very  easily  set  out 
on  the  ground,  as  directed  in  Art.  (90),  under  "  Platting,"  usipg 
a  rope  or  chain  for  compasses. 

(495)  The  content  and  base  being  given,  and  one  side  having 
to  make  a  given  angle,  as  B,  with  the  base 

AB,  the  length  of  the  side  BC  =^  ^  ^^^  . 
'  ^  AB .  sin.  B 

Example.     Eighi^y  acres  are  to  be  laid 

out  in  the  form  of  a  triangle,  on  a  base, 

AB,  of  sixty  chains,  bearing  N.  80°  W. 

the  bearing  of  the  side  BC  being  N.  70°  E.     Here  the  angle  B  is 

found  from  the  Bearings  (by  Art.  (243),  reversing  one  of  them) 

to  be  30°.     Hence  BC  =  53.33.     The  figure  is  on  a  scale  of  5C 

chains  to  1  inch  =  1 :  39600. 

Any  right-Hne  figure  may  be  laid  out  by  analogous  methods. 

(49(5)  To  lay  out  circles.  Multiply  the  given  content  by  7, 
divide  the  product  by  22,  ana  take  the  square  root  of  the  quotient 


CUAP.  I.] 


Laying  out  Land. 


S3& 


This  will  give  the  radius,  with  which  the  circle  can  be  described 
on  the  ground  with  a  rope  or  chain.  A  circle  containing  one  acre 
has  a  radius  of  178^  links.  A  circle  containing  a  quarter  of  an 
acre  will  have  a  radius  of  89  links. 


(497)  Town  lots.  House  lots  in  cities  are  usually  laid  off  as 
rectangles  of  25  feet  front  and  100  feet  depth,  variously  combined 
in  blocks.  Part  of  New- York  is  laid  out  in  blocks  200  feet  by 
800,  each  containing  64  lots,  and  separated  by  streets,  60  feet 
wide,  running  along  their  long  sides,  and  avenues,  100  feet  wide, 
on  their  short  sides.  The  eight  lots  on  each  short  side  of  the  block, 
front  on  the  avenues,  and  the  remaining  forty-eight  lots  front  on 
the  streets.  Such  a  block  covers  almost  precisely  3|  acres,  and 
17^  such  lots  about  make  an  acre.  But,  allowing  for  the  streets, 
land  laid  out  into  lots,  25  by  100,  arranged  as  above,  would  con- 
tain only  11.9,  or  not  quite  12  lots  per  acre. 

Lots  in  small  towns  and  villages  are  laid  out  of  greater  size  and 
less  uniformity.  50  feet  by  100  is  a  frequent  size  for  new  villages, 
the  blocks  being  200  feet  by  500,  each  therefore  containing  20  lots. 


(498)  Laud  sold  for  taxes.  A  case  occurring  in  the  State  of 
New- York  will  serve  as  an  application  of  the  modes  of  laying  out 
squares  and  rectangles.    Land  Fig-  343. 

on  which  taxes  are  unpaid  is 
sold  at  auction  to  the  loioest 
bidder ;  i.  e.  to  him  who  will 
accept  the  smallest  portion  of 
it  in  return  for  paying  the  taxes 
on  the  whole.  The  lot  in  ques- 
tion was  originally  the  east 
half  of  the  square  lot  ABCD, 
containing  500  acres.  At  a 
sale  for  taxes  in  1830,  70  acres 
were  bid  off,  and  this  area  was 
Bet  off  to  the  purchaser  in  a  square  lot,  from  the  north-east  comer. 
Required  the  side  of  the  square  in  links.  Again,  in  1834,  29 
acres  more  were  thus  sold,  to  be  set  off  in  a  strip  of  equal  width 


33 i  LAYIAG  OIT  AlVD  DIVIDING  UP  LA^D.      [part  xi 

around  the  square  previously  sold.  Required  the  width  of  this 
strip.  Once  more,  in  1839,  42  acres  more  were  sold,  to  be  set 
ofiF  around  the  preceding  piece.  Required  the  dimensions  of  this 
third  portion.  Tlie  answer  can  be  proved  by  calculating  if  the 
dimensions  of  the  remaining  rectangle  will  give  the  content  which 
it  should  have,  viz.  250  —  (70  +  29  +  42)  =  109  Acres. 
The  figure  is  on  a  scale  of  40  chains  to  1  inch  =  1 :  31680. 

(■199)  New  COHfltrieSt  The  operations  of  laying  out  land  for  the 
purposes  of  settlers,  are  required  on  a  large  Fcale  in  new  countries, 
in  combination  with  their  survey.  There  is  great  difficulty  in 
uniting  the  necessary  precision,  rapidity  and  cheapness.  "  Tri- 
angular Surveying"  will  ensure  the  first  of  these  qualities,  but  is 
deficient  in  the  last  two,  and  leaves  the  laying  out  of  lots  to  be 
subsequently  executed.  "  Compass  Surveying"  possesses  tlie  last 
two  qualities,  but  not  the  first.  The  United  States  system  for 
surveying  and  laying  out  the  Pubhc  Lands  admirably  combines  an 
accurate  determination  of  standard  lines  (Meridians  and  Parallels) 
with  a  cheap  and  rapid  subdivision  by  compass.  The  subject  is  so 
important  and  extensive  that  it  will  be  explained  by  itself  in 
Part  XII. 


CHAPTER  II. 


"V^ 


PARTING  OFF  LAND. 

(500)  It  is  often  required  to  part  off  from  a  field,  or  from  an 
mdefinite  space,  a  certain  number  of  acres  by  a  fence  or  other 
boundary  fine,  which  is  also  required  to  run  in  a  particular  direc- 
tion, to  start  from  a  certain  point,  or  to  fulfil  some  other  condition. 
The  various  cases  most  likely  to  occur  -svill  be  here  arranged 
according  to  these  conditions.  Both  graphical  and  numerical  me- 
thods will  generally  be  given.* 

*  The  given  lines  will  be  represented  by  fine  full  lines;  the  lines  of  constructioB 
by  broken  lines,  and  the  lines  of  the  result  by  heavy  full  lines. 


CHAP.  II.] 


Partins;  off  Land. 


335 


The  given  content  is  always  supposed  to  be  reduced  to  square 
chains  and  decimal  parts,  and  the  lines  to  be  in  chains  and  deci- 
mals. 

A.    Br  A  LiisrE  parallel  ro  a  side. 

(501)  To  part  off  a  rectau|?le.  If  the  sides  of  the  field  adja- 
cent to  the  given  side  make  right  angles  Avith  it,  the  figure  parted 
off  by  a  parallel  to  the  given  side  will  be  a  rectangle,  and  its 
breadth  will  equal  the  required  content  divided  by  that  side,  as  in 
Art.  (493). 

K  the  field  be  bounded  by  a  curved  or  zigzag  line  outside  of  the 
given  side,  find  the  content  between  these  irregular  lines  and  the 
given  straight  side,  by  the  method  of  offsets,  subtract  it  from  the 
content  required  to  be  parted  off,  and  proceed  with  the  remainder 
as  above.     The  same  directions  apply  to  the  subsequent  problems. 


(502)  To  part  off  a  parallelogram. 

the  given  side  be  parallel,  the 
figure  parted  off  will  be  a  parallel- 
ogram, and  its  perpendicular  width, 
CE,  will  be  obtained  as  above. 
The  length  of  one  of  the  parallel 

.,  ,^        CE  ABDC 

Bides,  as  AC  =  - — -  =  -r-rr — : r- 

sin.  A       AB  .  sm.  A 


If  the  sides  adjacent  to 

Fig.  344. 
C  D 


(503)  To  part  off  a  trapezoid.  When  the  sides  of  the  field 
adjacent  to  the  given  side  are  not  parallel,  the  figure  parted  off 
will  be  a  trapezoid. 

When  the  field  or  figure  is  given  on  the  ground,  or  on  a  plat, 
begin  as  if  the  sides  were  parallel,  Fig.  345. 

dividing  the  given  content  by  the 
base  AB.  The  quotient  will  be 
an  approximate  breadth,  CE,  or 
DF;  too  small  if  the  sides  con- 
verge, as  in  the  figure,  and  vice 
versa.  Measure  CD.  Calculate 
the  content  of  ABDC.     Divide  the  difference  of  it  and  the  required 


336 


LAYING  OUT  AND  DIVIDING  UP  LAND.      [part  xi. 


content  bj  CD.  Set  off  the  quotient  perpendicular  to  CD,  (m  this 
figure,  outside  of  it,)  and  it  will  give  a  new  line,  GH,  a  still  nearer 
approximation  to  that  desired.  The  operation  maj  be  repeated,  if 
found  necessary. 


(504)  "W^ien  the  field  is  given  by  Bearings,  de- 
duce from  them,  as  in  Art.  (243),  the  angles  at  A 
and  B.  The  required  sides  will  tlien  be  given  by 
tliese  formulas : 

2  X  ABCD  .  sin.  (A  +  B)\ 


Fig.  346. 


vathi>,h'on  CD=y(AB 


^AD  =  (AB  — CD)^ 


sin.  A 
sin.  B 


sin.  B 


sin.  (A  +  B) 
sin.  A 


BC  =  (AB  — CD)  .     ^.  ^^.. 
^  ■  sin.  (A  4-  B) 

VVlien  the  sides  AD  and  BC  diverge,  instead  of  converging,  as 
in  the  figure,  the  negative  term,  in  the  expression  for  CD,  becomes 
positive ;  and  in  the  expressions  for  both  AD  and  BC,  the  first 
factor  becomes  (CD  —  AB). 

The  perpendicular  breadth  of  the  trapezoid  =  AD  .  sin.  A ; 
or  =  BC  .  sin.  B. 

Exam]}le.  Let  AB  run  North,  six  chains ;  AD,  N.  80°  E. ; 
BC,  S.  60°  E.  Let  it  be  required  to  part  off  one  acre  by  a  fence 
parallel  to  AB.  Here  AB  =  6.00,  ABCD  =  10  square  chains, 
A  =  80°,B  =  60°.  Ans.  CD  =  4.57,  AD  =  1.92,  BC  =  2.18, 
and  the  breadth  =  1.89. 

The  figure  is  on  a  scale  of  4  chains  to  1  inch  =  1 :  3168. 

B.    By  a  line  perpendicular  to  a  side. 


(505)  To  part  off  a  triangle. 

When   the   field    is    given    on 


Let  FG  be  the  required  line, 
the  Fig-  347. 


ground,  or  on  a  plat,  at  any  point,  as 

D,  of  the  given  side  AB,  set  out  a 

"  guess  line,"  DE,  perpendicular  to 

AB,  and   calculate   the   content  of   ^'  d   r 

DEB.     Then  the  required  distance  BE,  from  the  angular  pomt 

to  the  foot  of  the  desired  perpendicular,  =  BD    l\—^ . 


CHAP,  n.] 


Parting  off  Land. 


337 


Example,     Let  BD  =  30  chains ;  ED  =  12  chains ;  and  the 
desired  area  =  24.8  acres.     Then  BF  =  35.22  chains. 
The  scale  of  the  ficnire  is  30  chains  to  1  inch  =  1 :  23760. 


BF  = 


=  7(- 


(506)  When  the  field  is  given  by  Bearings, 
5nd  the  angle  B  from  the  Bearings ;  then  is 
r2  X  BFGv 
tang.  B~/ ' 
Example.     Let  BA  bear  S.  75°  E.,  and  BC 
N.  60^  E.,  and  let  five  acres  be  required  to  be 
parted  oflf  from  the  field  by  a  perpendicular  to  BA.     Here  the 
angle  B  =  45%  and  BF  =  10.00  chams. 

The  scale  of  the  figure  is  20  chams  to  1  inch  =  1 :  15810. 


(507)  To  part  off  a  qnadrilatcral.  Produce  the  converging 
sides  to  meet  at  B.     Calculate  the  Fig.  369. 

content  of  the  triangle  HKB,  whe- 
ther on  the  ground  or  plat,  or  from 
Bearings.     Add  it  to  the  content 
of  the  quadrilateral  required  to  be   ^ 
parted  off,  and  it  wiU  give  that  of  the  triangle  FGB,  and  the  me- 
thod of  the  preceding  case  can  then  be  applied. 

(508)  To  part  off  any  figure.  If  the  field  be  very  irregularly 
shaped,  find  by  trial  any  line  ■wlii6h  will  part  oflf  a  httle  less  than 
the  required  area.  This  trial  line  vrill  represent  HK  in  the  pre- 
ceding figure,  and  the  problem  is  reduced  to  parting  oflf,  accord- 
ing to  the  required  condition  a  quadrilateral,  comprised  between 
the  trial  Ime,  two  sides  of  the  field,  and  the  required  line,  and  con- 
taining the  difference  between  the  required  content  and  that  parted 
off  by  the  trial-lmej 

C.      By   A   LIXE   RUNNING   IN   ANY   GIVEN   DIRECTION. 

(509)  To  part  off  a  triangle.  By  construction,  on  the  ground 
or  the  plat,  proceed  nearly  as  in  Art.  (505),  setting  out  a  line 
in  the  required  direction,  calculating  the  triangle  thus  formed,  and 
obtaining  BF  by  the  same  formula  as  in  that  Article. 

22 


338 


LAYIXG  OUT  AND  DIVIDING  UP  LAND.       [parxxi 


7J 
^ 

V 

^ 


~  '<-^ 


H^ 


(510)  If  the  field  be  given  by  Bearings,  find 
from  them  the  angles  CBA  and  GFB ;  then  is 
1/2  X  BF&  sin   (B_+j;)\ 
V\         siQ.  B  .  sin.  F  /■ 

Example.  Let  BA  bear  S.  30°  E. ;  BC, 
N.  80°  E. ;  and  a  fence  be  required  to  run, from 
some  point  in  BA,  a  due  North  course,  and  to 
part  off  one  acre.  Required  the  distance  from 
B  to  the  point  F,  whence  it  must  start.  Ayis. 
The  angle  B  =  70°,  and  F  =  30°.  Then  BF  = 
6.47. 

The  scale  of  Fig.  350  is  6  chains  to  1  inch  =  1 :  4752. 


(511)  To  part  off  a  quadrilateral.    Let  it  be  required  to  part 
off,  by  a  line  running  in  a  ^^g-  ^^i- 

given  direction,  a  quadrila- 
teral from  afield  in  which 
are  given  the  side  AB,  and 
the  directions  of  the  two 
other  sides  running  from  A 
and  from  B. 

On  the  ground  or  plat 
produce  the  two  converging 
sides  to  meet  at  some  point 
E.  Calculate  the  content 
of  the  triangle  ABE.  Measure  the  side  AE.  From  ABE  subtract 
the  area  to  be  cut  off,  and  the  remainder  will  be  the  content  of  the 
triangle  CDE.  From  A  set  out  a  line  AF  parallel  to  the  given 
direction.     Find  the  content  of  ABF.     Take  it  from  ABE,  and 

tAus  obtain  AFE.     Then  this  formula,  ED  =  AE  <  7^5^  will  fix 

V  FAE 

the  point  D,  since  AD  =  AE  —  ED. 


(512)  When  the  field  and  the  dividing  line  are  given  by  Bear 
'ings,  produce  the  sides  as  in  the  last  article.  Find  all  the  angles 
from  the  Bearings.  Calculate  the  content  of  the  triangle  ABE,  by 
the  formula  for  one  side   and   its  including  angles.     Take   the 


cuAP.  n.]  Parting  off  Land;  339 

desii-ed  content  from  this  to   obtain   CDE.     Calculate  the  side 

AE  =  AB^.  ThenisAD==AE-  /(^  ^  CDE  .  sin. DCEv ; 
sm,  E  V  \    sin.  E  .  sin.  CDE     / 

Example.  Let  DA  bear  S.  20^°  W. ;  AB,  N.  51^°  W.,  8.19 ; 
J*C.  N.  73^-°  E. ;  and  let  it  be  required  to  part  off  two  acres  bj  a 
fence,  DC,  runnmg  N.  45°  W.  Am.  ABE  =  32.50  sq.  chains ; 
whence  CDE  =  12.50  sq.  chs.  Also,  AE  =  8.37;  and  finally 
AD  =  8.37  —  5.49  ==  2.88  chains. 

The  scale  of  Fig.  351  is  5  chams  to  1  inch  =  1 :  3960. 

If  the  sum  of  the  angles  at  A  and  B  was  more  than  two  right 
angles,  the  point  E  would  He  on  the  other  side  of  AB.  The  neces- 
sary modifications  are  apparent. 

(513)  To  part  off  any  figures  Proceed  in  a  similar  manner  to 
that  described  in  Art.  (508),  by  getting  a  suitable  trial-hne,  pro- 
ducing the  sides  it  intersects,  and  then  applying  the  method  just 
given. 

D.      By   A   LINE   STARTING   FROM   A   GIVEN  POINT  IN   A   SIDE. 

(514)  To  part  off  a  triangle.  Let  it  be  required  to  cut  off 
from  a  corner  of  a  field  a  triangu-  Fig.  352. 

lar  space  of  given  content,  by  a 
line  starting  from  a  given  point 
on  one  of  the  sides,  A  in  the  figure, 
the  base,  AB,  of  the  desired  tri- 
angle being  thus  given.  If  the 
field  be  given  on  the  ground  or  on 
a  plat,  divide  the  given  content 
by  half  the  base,  and  the  quotient  will  be  the  height  of  the  tri- 
angle. Set  off  this  distance  from  any  point  of  AB,  perpendicular 
to  it,  as  from  A  to  C ;  ti-om  C  set  out  a  parallel  to  AB,  and  ita 
intersection  with  the  second  side,  as  at  D,  will  be  the  vertex  of  the 
required  triangle. 

Otherwise,  divide  the  required  content  by  half  of  the  perpendi- 
cular distance  from  A  to  BD,  and  the  quotient  wiU  be  BD. 

*  This  original  formula  is  very  convenient  for  logarithmic  computation. 


840  LAYING  OUT  AXD  DIVIDING  CP  LAND.      [part  xi 

(515)  If  the  field  be  given  by  the  Bearings  of  two  sides  and  the 
length  of  one  of  them,  deduce  the  angle  B  (Fig.  352)  from  the 

2  X  ABD 

Bearings,  as  in  Art.  (243).     Then  is  BD  =  -^3 — -. — ^. 
="  '  ^       ^  AJ3  .  sin.  B 

If  it  is  more  convenient  to  fix  the  point  D,  by  the  Second  Me- 
thod, Art.  (6),  that  of  rectangular  co-ordinates,  we  shall  have 
BE  =  BD  .  COS.  B  ;  and  ED  =  BD  .  sin.  B. 

The  Bearing  of  AD  is  obtained  from  the  angle  BAD ;  which  \9 

ED  ED  ,         -r,*-n 

known,  smce  — =  ^^— ^=tang.  BAD. 

Example.  Eighty  acres  are  to  be  set  off  from  a  corner  of  a 
Geld,  the  course  AB  being  N.  80^  W.,  sixty  chains  ;  and  the  Bear- 
ing of  BD  being  N.  70^  E.  Ans.  BD  =  53.33  ;  BE  =  46.19  : 
ED  =  26.67  ;  and  the  Bearing  of  AD,  N.  17°  23'  W. 

The  scale  of  Eig.  352  is  40  chains  to  1  inch  =  1 :  31680. 

2  ABD 
If  the  field  were  right  angled  at  B,  of  course  BD  =  — p5— . 

AB 

(516)  To  part  off  a  quadrilateral.  Imagine  the  two  converg- 
mg  sides  of  the  field  produced  to  meet,  as  in  Art.  (511).  Calcu- 
late the  content  of  the  triangle  thus  formed,  and  the  question  will 
then  be  reduced  to  the  one  explained  in  the  last  two  articles. 

(517)  To  part  off  any  figure.  Proceed  as  directed  in  Art.  (513). 
Otherwise,  proceed  as  follows. 

The  field  being  given  on  the  ground  or  on  a  plat,  find  on  which 
side  of  it  the  required  line  will  end,  by  drawing  or  running  "  guesa 
lines"  from  the  given  point  to  various  angles,  and  roughly  measur- 
ing the  content  thus  parted  off.  fig-  353. 
If,  as  in  the  figure,  A  being  the 
given  point,  the  guess  line  AD 
parts  off  less  than  the  required  coi 
tent,  and  AE  parts  off  more,  then 
the  desired  division  line  AZ  will 
end  in  the  side  DE.    Subtract  the 
area  parted  off  by  AD  from  the 
required  content,  and  the  difference  will  be  the  content  of  the  tri 


CHAP.  11.]  Parting  off  Land.  341 

angle  ADZ.  Divide  this  by  half  the  perpendicular  let  fall  from 
the  given  point  A  to  the  side  DE,  and  the  quotient  will  be  the  base, 
or  distance  from  D  to  Z. 

Or,  find  the  content  of  ADE  and  make  this  proportion  ;  ADE  : 
ADZ  : :  DE  :  DZ. 

(518)  The  field  being  given  by  Bearings  and  distances,  find 
as  before,  by  approximate  trials  on  the  plat,  or  otherwise,  which 
side  the  desired  line  of  division  will  terminate  in,  as  DE  in  the  last 
figure.  Draw  AD.  Find  the  Latitude  and  Departure  of  this 
line,  and  thence  its  length  and  Bearing,  as  in  Art.  (440).  Then 
calculate  the  area  of  the  space  this  line  parts  ofi",  ABCD  in  the 
figure,  by  the  usual  method,  explained  in  Part  III,  Chapter  VI. 
Subtract  this  area  from  that  required  to  be  cut  ofi",  and  the  remain- 
der will  be  the  area  of  the  triangle  ADZ.  Then,  as  in  Art.  (515), 
P^^_        2  ADZ 

AD  .  sin.  ADZ* 

This  problem  may  be  executed  without  any  other  Table  than  that 
of  Latitudes  and  Departures,  thus.  Find  the  Latitude  and  Depar- 
ture of  DA,  as  before,  the  area  of  the  space  ABCD,  and  thence 
the  content  of  ADZ.  Then  find  the  Latitude  and  Departure  of 
EA,  and  the  content  of  ADE.  Lastly,  make  this  proportion : 
ADE  :  ADZ  : :  DE  :  DZ.* 

Example.  In  the  field  ABCDE,  &c.,  part  of  which  is  shown 
in  Fig.  353,  (on  a  scale  of  4  chains  to  1  inch=  1 :  3168),  one 
acre  is  to  be  parted  off  on  the  west  side,  by  a  line  starting  from  the 
angle  A.  Required  the  distance  from  D  to  Z,  the  other  end  of 
this  dividing  Une.f 

The  only  courses  needed  are  these.  AB,  N.  53°  W.,  1.55 , 
BC,  N.  20^  E.,  2.00  ;  CD,  N.  53^°  E.,  1.32  ;  DE,  S.  57°  E.,  5.79. 
A  rough  measurement  will  at  once  shew  that  ABCD  is  less  than 
an  acre,  and  that  ABCDE  is  more  ;  hence  the  desired  line  will  fall 

*  The  problem  may  also  be  performed  by  making  the  side  on  which  the  divi 
sion  line  is  to  fall,  a  Meridian,  and  changing  the  Bearings  as  in  Art.  (244).  The 
ditference  of  the  new  Departures  will  be  the  Departure  of  the  Division  line.  Its 
position  can  tlieu  be  easily  determined,  by  calculations  resembling  those  in  Part 
VII,  Chapter  IV,  Arts.  (443),  &c. 

t  If  the  whole  field  has  been  surveyed  and  balanced,  the  balanced  Latitudes 
and  Departures  should  be  used.  We  will  here  suppose  the  survey  to  have  proved 
perfectly  correct. 


84: 


LAYING  OUT  AND  DIVIDING  UP  LAND,      [part  xi. 


on  DE.  The  Latitudes  and  Departures  of  AB,  BC  and  CD  are 
then  found.  From  them  the  course  AD  is  found  to  be  N.  8^  E., 
3.63.  The  content  of  ABCD  will  be  3.19  square  chains.  Sub. 
tracting  this  from  one  acre,  the  remainder,  6.81  sq,  chs.,  is  the  con- 
tent of  ADZ.  AP=  3.63  x  sin.  65°  =  3.29.  Dividing  ADZ 
bj  half  of  this,  we  obtain  DZ  =  4.14  chains. 

Bj  the  Second  Method,  the  Latitude  and  Departure  of  DA,  the 
area  of  ABCD,  and  of  ADZ,  being  found  as  before,  we  next  find 
the  Latitude  and  Departure  of  EA,  from  those  of  AD  and  DE, 
and  thence  the  area  of  ADE  =  9.53.  Lastly,  we  have  ..he  pro- 
portion 9.53  :  6.81  : :  5.79  :  DZ  =  4.14,  as  before.  • 

E.  By  a  line  passing  through  a  given  point  within  the  field. 


Let  P  be  a  point  within  a  field 
m'  b 


(519)  To  part  off  a  triangle. 

through  which  it  is  required  to 
run  a  line  so  as  to  part  off  from 
the  field,  a  given  area  in  the 
form  of  a  triangle. 

When  the  field  is  given  on  the 
ground  or  on  a  plat,  the  division 
can  be  made  by  construction, 
thus.  From  P  draw  PE,  paral- 
lel to  the  side  BC.  Divide  the 
given  area  by  half  of  the  perpen- 
dicular distance  from  P  to  AC, 
and  set  off  the  quotient  from  C 
to  G.  Bisect  GC  in  H.  On 
HE  describe  a  semi-circle.  On 
it  set  off  EK  =  EC.  Join  KH. 
Set  off  HL  =  HK.  The  line  LM,  drawn  from  L  through  P,  will 
be  the  division  line  required.*  If  HK  be  set  off  in  the  contrary 
direction,  it  will  fix  another  line  LTM',  meeting  CB  produced,  and 
thus  parting  off  another  triangle  of  the  required  content. 

Example.     Let  it  be  required  to  part  off  31.175  acres  by  a 
fjnce  passing  through  a  point  P,  the  distance  PD  of  P  from  the 

*  As  some  lines  in  the  figure  are  not  used  in  the  construction,  though  needed 
for  the  Demouslratiou,  the  stiidcnt  siiould  draw  it  liimself  to  a  large  scale. 


G 


CHAP.   11.] 


Parting  ofT  Land. 


343 


side  BC,  measured  parallel  to  AC,  being  6  chains,  and  DC  18 
chains.  The  angle  at  C  is  fixed  by  a  "tie-line"  AB  =  48.00^ 
BC  being  42.00,  and  CxV  being  30.00.  Ans.  CL  =  27.31 
chains,  or  CL'  =  7.69  chains. 

The  figure  is  on  a  scale  of  20  chains  to  1  inch  =  1  :  15840. 


(520)  If  the  angle  of  the  field 
and  the  position  of  the  point  P  are 
given  by  Bearings  or  angles,  proceed 
thus.  Find  the  perpendicular  dis- 
tances, PQ  and  PB,  from  the  given 
point  to  the  sides,  by  the  formulas 
PQ  =  PC  .  sin.  PCQ ;  and  PR  = 
PC  .  sin.  PCR.  Let  PQ  =  q,  PR 
"^p,  and  the  required  content  =  c. 


Then  CL  = 


P       V  \1 


2qc 


p       V   yp"       sin.  LCM/ 
Example.     Let  the  angle  LCM  =  82^.     Let  it  be  required  to 
part  off  the  same  area  as  in  the  preceding  example.     Let  PC  = 


19.75,  PCQ  =  17^   301',   PCR  =  64' 


291'. 


Requii-ed   CL. 


Ans.  PQ  =  5.94,  PR  =17.82,  and  therefore,  by  the  formula, 
CL  =  27.31,  or  CL'  =  7.69  ;  corresponding  to  the  graphical 
solution.     The  figure  is  on  the  same  scale. 

If  the  given  pomt  were  without  the  field,  the  division  line  could 
be  determined  in  a  similar  manner. 


(521)  To  part  off  a  quadrilateral>     Conceive  the  two  sides  of 
the  field  -which  the  division  line  will  intersect,  pig.  356. 

DA  and  CB,  produced  till  they  meet  at  a 
point  G,  not  shown  in  the  figure.  Calculate 
the  triangle  thus  formed  outside  of  the  field. 
Its  area  increased  by  the  required  area, 
will  be  that  of  the  triangle  EFG.  Then  the 
problem  is  identical  with  that  in  the  last 
article.  The  following  example  is  that 
given  m  Gummere's  Surveying.  The  figure 
represents  it  on  a  scale  of  20  chains  to  1  inch  =  1 :  15840. 


844 


LAYING  OUT  AND  DIVIDING  UP  LAND.      [part  xi 


Example.  A  field  is  bounded  thus:  N.  14^  W.,  15  20: 
N.  701°  E.,  20.43  ;  S.  6°  E.,  22.79  ;  N.  86i<^  W.,  18.00.  A 
spring  within  it  bears  from  the  second  corner  S.  75°  E.,  7.90.  It 
is  required  to  cut  off  10  acres  from  the  West  side  of  the  field  by  a 
straight  fence  through  the  spring.  How  far  will  it  be  from  the 
first  corner  to  the  point  at  which  the  division  fence  meets  the  fourth 
side?     Ayis.  4.6357  chains. 


Let  it  be  required  to  part  off 

Fig.  357. 

B  D 


«..— -I)' 


(522)  To  part  off  any  figure 

from  a  field  a  certain  area  by 

a  line  passing  through  a  given 

point  P  within  the  field.     Run 

a  guess-line  AB   through  P. 

Calculate  the   area  which  it 

parts  off.     Call  the  difference 

between  it  and  the  required 

area  =  d.     Let   CD   be   the 

desired  line  of  division,  and 

let  P  represent  the  angle,  APC  or  BPD,  which  it  makes  with  the 

given  line.     Obtain  the  angles  PAC  =  A,  and  PBD  =  B,  either 

by  measurement,  or  by  deduction  from  Bearings.     Measure  PA 

and  PB.     Then  the  desired  angle  P  will  be  given  by  the  following 

formula. 

P  =  — |(cot.  A+  cot.  B  — ^^'^^^')jz 
pAP^  .  cot.  B  —  BP^  .  cot.  A 


Cot. 


7 


L 


2d 


cot.  A  .  cot.  B  + 


f  cot.  A  +  cot  B  — 


AP2— BP^ 

'Id 


n 


If  the  guess  line  be  run  so  as  to  be  perpendicular  to  one  of  the 
Bides  of  the  field,  at  A,  for  example,  the  preceding  expression 
reduces  to  the  following  simpler  form. 


Cot.  P 


=  —  l  (cot.  B  — 


) 


vc 


AP2    cot.  B 
T5 


+  I  (cot.  B 


AP2  — BP2 

2  d 

_  AP2  — BP 
¥d 


± 


y\ 


CHAP,  u.] 


Partiui?  off  Land. 


345 


Example.  It  was  required  to  cut  off  from  a  field  twelve  acres 
by  a  line  passing  through  a  spring,  P.  A  guess-line,  AB,  was  run 
making  an  angle  with  one  side  of  the  field,  at  A,  of  55'^,  and  with 
the  opposite  side,  at  B,  of  Sl*^.  The  area  thus  cut  oflF  was  found 
to  be  13.10  acres.  From  the  spring  to  A  was  9.30  chains,  and  to 
B  3.30  chains.  Required  the  angle  which  the  required  line,  CD, 
must  make  with  the  guess  line,  AB,  at  P.  Ans.  20^  45' ;  or 
—  86°  25'.     The  heavy  broken  Ime,  CD',  shows  the  latter. 

The  scale  of  the  figure  is  10  chains  to  1  inch  =  1  :  7920. 

If  the  given  point  were  outside  of  the  field,  the  calculations  would 
be  similar 

F.    By  the  shortest  possible  line. 

(523)  To  part  off  a  triangle.     Let  it  be  required  to  part  off  a 
triangular  space,  BDE,  of  given  content,  from  the         Fig.  358. 
corner  of  a  field,  ABC,  by  the  shortest  possible 
Une,  DE. 

From  B  set  off  BD  and  BE  each  equal  to 

//2  BDE\  ^    ^^  j.^^  j^j,  ^j^^g  obtained  will  be 
V   \  sin.  B  / 
perpendicular  to  the  line,  BF,  which  bisects  the  an- 

de  B.   The  length  of  DE  =  -^^ ,-=; -. 

^  °  COS.  ^  B 

Example.  Let  it  be  required  to  part  off  1.3  acre  from  the 
comer  of  a  field,  the  angle,  B,  being  30^.  Ans.  BD  =  BE  = 
7.21;  and  DE  =  3.73. 

The  scale  of  the  fio-ure  is  10  chams  to  1  iach  =1 :  7920. 


G.     Land  of  variable  value. 

(524)  Let  the  figure  represent  a  field  in  which 
the  land  is  of  two  qualities  and  values,  divided  by 
the  "  quahty  line"  EF.  It  is  required  to  part  off 
from  it  a  quantity  of  land  worth  a  certain  sum,  by 
a  straight  fence  parallel  to  AB. 

Multiply  the  value  per  acre  of  each  part  by  its 
length  (in  chains)  on  the  line  AB,  add  the  pro- 
ducts, multiply  the  value  to  be  set  off  by  10,  divide 


Fi£ 


359. 
C 


S46  LAYING  OUT  AND  DIVIDING  UP  LAND.       [partxi 

bj  the  above  sum,  and  the  quotient  will  be  the  desired  breadth,  BC 
or  AD,  in  chains. 

Example.  Let  the  land  on  one  side  of  EF  be  worth  $200  pei 
acre,  and  on  the  other  side  $100.  Let  the  length  of  the  former, 
BE,  be  10  chains,  and  EA  be  30  chains.  It  is  requii'ed  to  part 
off  a  quantity  of  land  worth  $7500.  Ans.  The  width  of  the 
desired  strip  will  be  15  chains. 

The  scale  of  the  figure  is  40  chains  to  1  inch  =  1 :  31680. 

K  the  "  quahtj  hne"  be  not  perpendicular  to  AB,  it  may  be 
made  so  by  "  giving  and  taking,"  as  in  Art.  (124),  or  as  in  the 
article  following  this  one. 

The  same  method  may  be  applied  to  land  of  any  number  of 
different  qualities ;  and  a  combuiation  of  this  method  with  the  pre- 
ceding problems  will  solve  any  case  which  may  occur. 

H.     Straightening  crooked  fences. 

(525)  It  is  often  required  to  substitute  a  straight  fence  for  a 
crooked  one,  sc  that  the  former  shall  part  off  precisely  the  same 
quantity  of  land  as  did  the  latter.  This  can  be  done  on  a  plat  by 
the  method  given  in  Art.  (83),  by  which   the  irregular  figure 

Fig.  360. 


L 


^ 


6^  -^ 

1...2...3...4...5  is  reduced  to  the  equivalent  triangle  1... 5. ..3',  and 
the  straight  line  5... 3'  therefore  parts  off  the  same  quantity  of  land 
on  either  side  as  did  the  crooked  one.  The  distance  from  1  to  3', 
as  found  on  the  plat,  can  then  be  set  out  on  the  ground  and  the 
straight  fence  be  then  ranged  from  3'  to  5 

The  work  may  be  done  on  the  ground  more  accurately  by  run- 
ning a  guess  line,  AC,  Fig.  361,  across  the  bends  of  the  fence  which 
crooks  from  A  to  B,  measuring  offsets  to  the  bends  on  each  side 
of  the  guess  line,  and  calculating  their  content.  If  the  sums  of 
these  areas  on  each  side  of  AC  chanced  to  be  equal,  that  would  be 
the  line  desired ;  but  if,  as  in  the  figure,  it  passes  too  far  on  one 


cnAP  III.] 


Dividing  up  Land. 

Fig.  361. 


347 


Bide,  divide  the  difference  of  the  areas  by  half  of  AC,  and  set  it 
off  at  right  angles  to  AC,  from  A  to  D.  DC  -will  then  be  a  line 
parting  off  the  same  quantity  of  land  as  did  the  crooked  fence.  If 
the  fence  at  A  was  not  perpendicular  to  AC,  but  oblique,  as  AE, 
then  from  D  run  a  parallel  to  AC,  meeting  the  fence  at  E,  and  EC 
will  be  the  required  line. 


CHAPTER  III. 


DIVIDING  UP  LAND. 

(526)  Most  of  the  problems  for  "  Dividing  up"  land  may  be 
brought  under  the  cases  in  the  preceding  chapter,  by  regarding 
one  of  the  portions  into  which  the  figure  is  to  be  divided,  as  an 
area  to  be  "  Parted  off"  from  it.  Many  of  them,  however,  can 
be  most  neatly  executed  by  considering  them  as  independent  pro- 
blems, and  this  will  be  here  done.  They  will  be  arranged,  firjtly, 
according  to  the  simphcity  of  the  figure  to  be  divided  up,  and  then 
sub-arranged,  as  in  the  leading  arrangement  of  Chapter  II,  accord- 
ing to  the  manner  of  the  division. 


DIVISION  OF  TRIANGLES. 

(527)  By  lines  parallel  to  a  side.  Sup- 
pose that  the  triangle  ABC  is  to  be  divided  into 
two  equivalent  parts  by  a  line  parallel  to  AC. 
The  desired  point,  D,  from  which  this  line  is  to 
Btart,  will  be  obtained  by  measuring  BD  = 


Fig.  362 


AB 


^h 


So,  too,  E  is  fixed  by  BE  =  BC  Vh- 


S48 


LATL\G  01 T  AND  DIVIDING  IP  L\ND,      [part  xi 


Generally,  to  divide  the  triangle  into  two  parts,  BDE  and  AGED 
which  shall  have  to  each  other  a  ratio  ==m  :  n,  we  have  BD  =» 

AB  J^H-. 

This  may  be  constructed   thus.     Describe  a  Fig-  363, 

semicircle  on  AB  as  a  diameter.     From  B  set 


offBF  =  — 1—  .BA. 

m  +  n 


Fig.  364. 


At  F  erect  a  perpendi- 
cular meeting  the  semicircle  at  G.     Set  off  BG 
from  B  to  D.     D  is  the  starting  point  of  the  divi- 
sion line  required.     In  the  figure,  the  two  parts  are  as  2  to  3,  and 
BF  is  therefore  =  f  BA. 

To  divide  the  triangle  ABC  into  five 
equivalent  parts,  we  should  have,  similarly, 
BD=AB  Vi;  BD'  =  AB  Vf;  BD" 
=  AB  Vf;  BD"'  =  AB  V|. 

The  same  method  will  divide  the  trian- 
gle into  any  desired  number  of  parts  hav- 
ing any  ratios  to  each  other. 


Suppose   that   ABC 

Fig.  3G5. 
B 


(528)  By  Imes  perpendicular  to  a  side 

is  to  be  di\ided  into  two  parts  having 
a  ratio  =  m  :  ?i,  by  a  hue  perpendicular 
to  AC.  Let  EF  be  the  dividing  luie 
whose  position  is  reqviired.  Let  BD 
be  a  perpendicular  let  fall  from  B  to 

AC.    ThenisAE=:^/(AC  X  AD  X 

AFE  :  EFBC  : :  m  :  ti  : :  1  :  2. 

If  the  triangle  had  to  be  divided  into  two  equivalent  parts,  the 
above  expression  would  become  AE  =  y/(\  AC  x  AD). 

(529)  By  lines  running  in  any  given  direction.  Let  a  triangle, 
ABC,  be  given  to  be  divided  into  two  parts,  having  a  ratio  =  m  :  w, 
by  a  line  making  a  given  angle  with  a  side.     Part  off,  as  in  Art 


m  +  n 


I       In  this  figure, 


(509)  or  C5I0),  Fig.  350,  an  area  BFG  = 


m 
m  A-  n 


•  ABC. 


CHAP.  III.] 


DiTiding  up  Land. 


?49 


(530)  By  lines  starting?  from  an  angle.     Divide  the  side  oppo 
site  to  the  given  angle  into  the  required  num-  Fi?.  366. 

ber  of  parts,  and  draw  lines  from  the  angle  to 
the  points  of  division.  In  the  figure  the  tri- 
angle is  represented  as  being  thus  divided  into 
two  equivalent  parts. 

If  the  triangle  were  required  to  be  divided  into  two  parts,  havin» 

to  each  other  a  ratio  —  m  :  n,  we  should  have  AD  =  AC 


m  -\-  n 


and  DC  =  AC  -—  — 
m  +  n 

If'  the  triangle  had  to  be  divided  into  three 
parts  which  should  be  to  each  other  : :  ??i :  n  :p, 

we   should   have   AD  =  AC 


=  AC 


367. 


,  and  EC  =  AC 


m  -{■  n  ■}-  p  m  +  n  ■{■  p' 

Suppose  that  a  triangular  field  ABC,  had  to  be  di\dded  among 
five  men,  two  of  them  to  have  a  quarter  each,  and  three  of  them 
each  a  sixth.  Divide  AC  into  two  equal  parts,  one  of  these  again 
mto  two  equal  parts,  and  the  other  one  into  three  equal  parts. 
Run  the  lines  from  the  four  points  thus  obtained  to  the  angle  B. 


(531)  By  lines  starting  from  a  point  in  a  side;    Suppose  that 
the  triangle  ABC  is  to  be  divided  into  ttvo  Fig.  368 

equivalent  parts  bj  a  hne  starting  from  a  point 
D  in  the  side  AC.  Take  a  pomt  E  in  the 
middle  of  AC.  Join  BD,  and  from  E  draw  a 
parallel  to  it,  meeting  AB  in  F.  DF  will  be 
the  dividing  hne  required. 

The  point  F  will  be  most  easily  obtamed  on  the  ground  by  the 
proportion  AD  :  AB  : :  AE  =  i  AC  :  AF. 

The  altitude  of  AFD  of  course  equals  ^  ABC  -^  ^  AD. 

If  the  triangle  is  to  be  divided  into  two  parts  havdng  any  other 
ratio  to  each  other,  divide  AC  in  that  ratio,  and  then  proceed  aa 

before.      Let  this  ratio  =  m  :  7i,  then  AF  = 


AD 


m  -{-  n 


350 


LAYING  OUT  AND  DIVIDING  UP  LAND.      [partxi 


(532)  Next  suppose  that  the  trian-  ^^s-  369. 
gle  ABC  is  to  be  divided  into  three 
'equivalent  parts,  meeting  at  D.     The 
altitudes,  EF  and  GH,  of  the  parts 
IDE  and  DCG,  will  be  obtained  hj  jn:  i-  d     x  h 
iividing  ^  ABC,  by  half  of  the  respective  bases  AD  and  DC. 

If  one  of  these  quotients  gives  an  altitude  greater  than  that  of  the 
triangle  ABC,  it  wlII  shew  that  the  two  lines  DE  and  DG  would 
both  cut  the  same  side,  as  in  Fig.  370,  in 
wrhich  EF  is  obtained  as  above,  and  GH  = 
I  ABC  -^  I  AD. 

In  practice  it  is  more  convenient  to  de- 
termine the  points  F  and  G,  by  these 
proportions ;  r       h  k     n 

BK  :  AK  : :  EF  :  AF;  and  BK  :  AK  : :  GH  :  AH. 

The  division  of  a  triangle  into  a  greater  number  of  parts,  having 
any  ratios,  may  be  effected  in  a  similar  manner. 

(533)  This  problem  admits  of  a  more  elegant  solution,  analogous 
to  that  given  for  the  division  into  two 
parts,  graphically.  Divide  AC  into 
three  equal  parts  at  L  and  M.  Join 
BD,  and  from  L  and  M  draw  paral- 
lels to  it,  meeting  AB  and  BC  in  E  "  id  m 
and  G.  Draw  ED  and  GD,  which  will  be  the  desired  lines  of 
division.     The  figure  is  the  same  triangle  as  Fig.  369. 

The  points  E  and  G  can  be  obtained  on  the  ground  by  measur- 
ing AD  and  AB,  and  making  the  proportion  AD  :  AB  : ;  |  AC  :  AE. 
The  point  G  is  similarly  obtained. 

The  same  method  will  divide  a  triangle  into  a  greater  number 
of  parts. 

(534)  To  divide  a  triangle  into  four  equivalent  triangles  by 
lines  terminating  in  the  sides,  is  very 
easy.  From  D,  the  middle  point  of  AB, 
draw  DE  parallel  to  AC,  and  from  F, 
the  middle  of  AC,  draw  FD  and  FE. 
The  problem  is  now  solved. 


Fig.  372. 


CHAP.  HI.] 


Dividing  up  Land. 


351 


(535)  By  lines  passing  through  a  point  within  the  triangle. 

Let  D  be  a  given  point  (such  as  a  well,  ^^g-  •'''3. 

&c.)  within  a  triangular  field  ABC,  from 

which  fences  are  to  run  so  as  to  divide 

the  triangle   into   tivo    equivalent  parts. 

Join  AD.     Take  E  m  the  middle  of  BC,    a- 

and  from  it  draw  a  parallel  to  DA,  meeting  AC  in  F.     EDF  is 

tlie  fence  required. 

(536)  If  it  be  required  to  di- 
vide a  triangle  into  two  equiva- 
lent parts  by  a  straight  line  pass- 
ing through  a  point  within  it,  pro- 
ceed thus.  Let  P  be  the  given 
point.  From  P  draw  PD  paral- 
lel to  AC,  and  PE  parallel  to  BC. 
Bisect  AC  at  F.  JoinFB.  From 
B  draw  BG  parallel  to  DF.  Then 
bisect  GC  m  H.  On  HE  de- 
scribe a  semicircle.  On  it  set  off 
EK  =  EC.  Join  KH.  Set  off 
HL  =  HK.  The  hue  LM  drawn 
from  L,  through  P,  will  be  the 
division  line  required. 

This  figure  is  the  same  as  that  of  Art.  (519).  The  triangle 
ABC  contains  62.35  acres,  and  the  distance  CL  =  27.81  chains, 
as  ill  the  example  in  that  article. 

(537)  Next  suppose  that  the  trian- 
gle ABC  is  to  be  divided  into  three 
equivalent  parts  by  lines  starting  from 
a  point  D,  within  the  triangle,  given  by 
the  rectangular  co-ordinates  AE  and 
and  ED.  Let  ED  be  one  of  the  lines  ^^~^  kt: 
of  division,  and  F  and  G  the  other  points  required.  The  point  F 
will  be  determined  if  AH  is  known  ;  AH  and  HF  being  its  rectan- 
gular co-ordinates.     From  B  let  fall  the  perpendicular  BK  on  AC. 


352 


LAYING  OUT  AND  DIVIDING  UP  LAND.       [part  xi. 


Then  is  AH  =  AKgABC^AExEI))  ^^^^ 

men  is  i^xi         aE  X  BK  —  ED  x  AK  ^ 

other  point,  G,  is  determined  in  a  similar  manner. 


E  li 


(538)  Let  DB,  instead  of  DE,  Fig.  376 

be  one  of  the  required  lines  of 
division.  Divide  ^  ABC  by  half 
of  the  perpendicular  DH,  let  fall 
from  D  to  AB,  and  the  quotient 
•will  be  the  distance  BF.  To  find 
G,  if,  as  in  this  figure,  the  trian-  ^^ 
gle  BDC  (=  BC  X  i  DK)  is  less  than  \  ABC,  divide  the  excess 
of  the  latter  (-which  will  be  CDG)  by  \  DE,  and  the  quotient  -will 
be  CG. 

Example.  Let  AB  =  30.00 ;  BC  =  45.00  ;  CA  =  50.00. 
Let  the  perpendiculars  from  D  to  the  sides  be  these  ;  DE  =  10.00  ; 
DH  =  20.00 ;  DK  =  5.17|.  The  content  of  the  triangle  ABC 
will  be  666.6  square  chams.  Each  of  the  small  triangles  must 
therefore  contain  222.2  sq.  chs.,  BD  bemg  one  division  line.  "We 
shall  therefore' have  BF  =  222.2  ^  ^  DH  =  22.2  chains.  BDC 
=  45  X  ^  X  5.17|  =  116.4  sq.  chs.,  not  enough  for  a  second  por- 
tion, but  leavmg  105.8  sq.  chs.  for  CDG ;  whence  CG  =  21.16 
chs.  To  prove  the  work,  calculate  the  content  of  the  remaining 
portion,  GDFA.  We  shaU  find  DGA  =  144.2  sq.  chs.,  and  ADF 
=  78.0  sq.  chs.,  making  together  222.2  sq.  chs.^  as  required. 

The  scale  of  Fig.  376  is  30  chains  to  1  inch  =  1  :  23760. 


(539)  The  preceding  case  may 
be  also  solved  graphically,  thus. 
TakeCL  =  ^AC.  Join  DL,  and 
from  B  draw  BG  parallel  to  DL. 
Join  DG.  It  will  be  a  second  line 
of  division.  Then  take  a  point,  / 
M,  in  the  middle  of  BG,  and  from  A'^ 


Fig.  377. 


0  L 


it  draw  a  line,  MF,  parallel  to  DA.  DF  will  be  the  third  line  of 
division.  This  method  is  neater  on  paper  than  the  preceding ;  but 
less  convenient  on  the  gi'ound. 


fHAP.  III.] 


Dividing  np  Land. 


353 


(540)  Let  it  be  required  to  divide  Fig.  378. 

the  triangle  ABC  into  three  equiva- 
lent triangles,  by  lines  drawn  from 
the  three  angular  points  to  some  im- 
known  point  within  the  triangle.  This 
point  is  now  to  be  found.  On  any  ^^ 
side,  as  AB.  take  AD  =  ^  AB.  From  D  draw  DE  parallel  to 
AC.     The  middle,  F,  of  DE,  is  the  pomt  required. 

If  the  three  small  triangles  are  not  to  be  equivalent,  but  are  to 
have  to  each  other  the  ratios ::  m:  n:p,  ^^s-  379. 
divide  a  side,  AB,  into  parts  having 
these  ratios,  and  through  each  point 
of  division,  D,  E,  draw  a  parallel  to 
the  side  nearest  to  it.  The  intersec- 
tion of  these  parallels,  in  F,  is   the    A'*=== — ^= ^==^C 

point  required.     In  the  figure  the  parts  ACF,  ABF,  BCF,  are  as 
2:3:4. 


(d4I)  Let  it  be  required  to  find 
the  position  of  a  point,  D,  situated 
within  a  given  ti'iangle,  ABC,  and 
equally  distant  from  the  points  A,  B, 
C ;  and  to  determine  the  ratios  to 
each  other  of  the  three  triangles  into 
which  the  given  triangle  is  divided. 

By  construction,  find  the  centre  of  the  circle  passing  through 
A,  B,  C.     This  will  be  the  required  point. 

By  calculation,  the  distance  DA  =  DB  =  DC  = 


AB  X  BC  X  CA 


4  X  area  ABC 

The  three  small  triangles  will  be  to  each  other  as  the  sines  of  their 
angles  at  D ;  i.  e.  ADB  :  ADC  :  BDC  : :  sin.  ADB  :  sin.  ADC  : 
sin.  BDC.  These  angles  are  readily  found,  since  the  sine  of  half 
of  each  of  them  equals  the  opposite  side  divided  by  twice  one  of 
the  equal  distances. 


23 


354  LAYING  OUT  Ai\D  DIVIDING  UP  LAND.      [part  xi 

(512)  By  the  shortest  possible  line.    Let  it  be        Fig.  38i. 
required  to  divide  the  triangle  ABC  bj  the  short 
est  possible  line,  DE,  into  two  parts,  which  shall 
be  to  each  other  ::  m  :  7i;  or  DBE  :  ABC  : :  m 

:  m  +  n. 

From  the  smallest  angle,  B,  of  the  triangle, 
measure  along  the  sides,  BA  and  BC,  a  distance 

BD  =  BE=,/(^-x  ABxBc).    DEisthe 

line  required.     It  is  perpendicular  to  the  line  BF  which  bisecta 

the  angle  ABC;  and  it  is  =   ^^\  ^  ,/(     ^      x  AB  x  BCl- 


COS. 


iB 


DIVISION    OF    RECTANGLES. 

(543)  By  lines  parallel  to  a  side.  Divide  two  opposite  sides 
into  the  required  number  of  parts,  either  equal  or  in  any  given 
ratio  to  each  other,  and  the  lines  joining  the  points  of  division  will 
be  the  hnes  desired. 

The  same  method  is  applicable  to  any  parallelogram. 

Example.  A  rectangular  field 
ABCD,  measuring  15.00  chains 
by  8.00,  is  bought  by  three  men, 
who  pay  respectively  $300,  $400 
and  $500.  It  is  to  be  divided 
among  them  in  that  proportion. 
Aris.  The  portion  of  the  first, 
AEE'B,  is  obtained  by  making  the  proportion  300  +  400  4-  500  : 
300  : :  15.00  :  AE  =  3.75.  EF  is  in  Uke  manner  found  to  be 
5.00  ;  and  FD  ^  6.25.  BE'  is  made  equal  to  AE ;  E'F  to  EF ; 
and  FC  to  FD.  Fences  from  E  to  E',  and  from  F  to  F',  will 
divide  the  land  as  required. 

The  scale  of  the  figure  is  10  chains  to  1  inch  =  1 :  7920. 

The  other  modes  of  dividmg  up  rectangles  will  be  given  undei 
the  head  of  "  Quadrilaterals,"  Art.  (548),  &c. 


CHAP.  III.] 


DiTidiug  up  Laud. 


35» 


DIVISION    OF    TRAPEZOIDS. 
(544)  By  Hues  parallel  to  the  bases.     Given  the  bases  and  a 


third  side  of  the  trapezoid,  ABCD,  to  be 
divided  into  two  parts,  such  that  BCFE  : 
EFDA  : :  m  :  n. 

The  length  of  the  desired  dividino;  line. 


EF 


=y( 


m  X  AD^  +  n  X  BC-^ 


The  distance  BE 


m  -\-  11  f 

_  AB  (EF  — BC) 


AD  — BC 

Example.  Let  AD  =  30  chains  ;  BC  = 
20  chs.  ;  and  AB  =  54^  chs. ;  and  the  parts 
to  be  as  1  to  2 ;  required  EF  and  BE. 
Ans.     EF  =  23.80  ;  and  BE  =  20.65. 

The  figure  is  on  a  scale  of  30  chains  to  1 
inch  =  1 :  23760. 


Fis  383. 


(545)  Given  the  bases  of  a  trapezoid,  and  the  perpendicular 
distance,  BH,  between  them ;  it  is  required  to  divide  it  as  before, 
and  to  find  EF,  and  the  altitude,  BG,  of  one  of  the  parts.     Let 


BCFE  :  EFDxV  : :  m  :  n.     Then  BG  = 


BCxBH 


/L 


AD— BC 


-.+ 


\  l_m  -f-  11 
EF  =  BC  +  BG  X 


2  X  ABCD  X  BH  . 

X   TT^ TTT^ 1- 


AD  — BC 
AD  — BC 


/BC  X  BH\2-i 
\AD  — BC/  J' 


BH 

Example.  Let  AD  =  30.00  ;  BC  =  20.00  ;  BH  =  54.00  ; 
and  the  two  parts  to  be  to  each  other  : :  46  :  89. 

The  above  data  give  the  content  of  ABCD  =  1350  square 
chains.  Substituting  these  numbers  in  the  above  formula,  vre  obtain 
BG  =  20.96,  and  EF  =  23.88. 


(546)  By  lines  starting  from  points  in  a  side.  To  divide  a 
trapezoid  into  parts  equivalent,  or  having  any  ratios,  divide  ^'ta 
parallel  sides  in  the  same  ratios,  and  join  the  corresponding  points. 


356 


LATI\G  OUT  AXD  DIVIDING  UP  LAXD.      [pakt  xi 


If  it  be  also  required  that  the  division  lines  shall  start  frou) 
given  points  on  a  side,  proceed 
thus.     Let  it  be  required  to 
divide   the   trapezoid   ABCD 
into  three  equivalent  parts  by 
fences  starting  from  P  and  Q 
Divide  the  trapezoid,  as  above 
directed,  into  three  equivalent    A- 
trapezoids  by  the  lines  EF  and  GH.     These  three  trapezoids  mu 
now  be  transformed,  thus.     Join  EP,  and  from  F  draw  FR  paral 
lei  to  it..    Join  PR,  and  it  will  be  one  of  the  didsion  lines  required. 

The  other  division  line,  QS,  is  obtained  similarly. 

(547)  Other  cases.  For  other  cases  :f  di\dding  trapcwids, 
apply  those  for  quadrilaterals  in  general,  given  in  the  following 
articles.* 

DIVISION    OF    QUADRILATERALS. 

(548)  By  lines  parallel  to  a  side.  Let  ABCD  be  a  quadrila 
teral  which  it  is  required  to 
divide,  by  a  line  EF,  paral- 
lel to  AD,  into  two  parts, 
BEFC  and  EFDA,  which 
shall  be  to  each  other  as 
m  :  n.  Prolong  AB  and  CD 
to  intersect  in  G.  Let  a  be 
the  area  of  the  triangle 
ADG,  obtained  by  any  me- 
thod, graphical  or  trigono- 
metrical, and  a'  =  the  area  "^^  b'  H  c 
of  the  triangle  BCG,  obtained  by  subtracting  the  area  of  the  given 
quadrilateral  from  that  of  the  triangle  ADG.     Then  GK  =  GH 

: — I .     Having  measured  this  length  of  GK  from  G  on 

{m  +  n)  af  °  =" 

GH,  set  off  at  K  a  perpendicular  to  GK,  and  it  will  be  the  required 

line  of  division. 


Fig.  385. 
G 


/(; 


•  If  a  line  be  drawn  joining  the  middle  points  of  the  parallel  bases  of  a  trape 
zcid,  any  line  drawn  through  the  middle  of  the  first  line,  and  meeting  the  paral 
le";  bases,  will  divide  the  trapezoid  into  two  equiyalent  parts. 


CHAP.  Ill]  Dividing  up  Landi  357 

Othenvise,  take  GE  =  GA.  /I — I  ;    and  from  E  run 

'  y\{m  +  n)af' 

a  pai'allel  to  xlD, 

If  the  tv^o  parts  of  the  quadrilateral  were  to  be  equivalent,  on  =>?, 

and  we  have  GK  =  GH  ,/(— ;t — -I  ;    and  consequently  GE  to 

GA  in  the  same  ratio. 

Example.  Let  a  quadrilateral,  ABCD,  be  required  to  be  thus 
livided,  and  let  its  angles,  B  and  C,  be  given  bj  rectangular  co-ordi- 
uates,  y\z  :  AB''  =  6.00 ;  B'B  =  9.00 ;  DC  =  8.00 ;  C'C  =  13.00 ; 
B'C  =  24.00.  Here  GH  is  readily  found  to  be  29.64  ;  ADG  = 
563.16  sqviare  chains  ;  and  BGC  =  220.16  square  chains.  Hence, 
by  the  formula,  GK  =  24.72  ;  whence  KH  =  GH  —  GK  =  4.92  ; 
and  the  abscissas  for  the  points  E  and  F  can  be  obtained  by  a 
simple  proportion. 

The  scale  of  the  figure  is  20  chains  to  1  inch  =  1 :  15840. 

If  the  quadrilateral  be  given  by  Bearings,  part  off  the  desired 

area  =  — - —  .  ABCD,  by  the  formulas  of  Art.  (504). 
m  -\-  n 

Suppose  now  that  a  quad-  Fig.  386. 

rilateral,  ABCD,  is  to  be  di-  Q 

vided  into  p  equivalent  parts,  ..-''     \ 

by  Unes  parallel  to  AD. 
Measure,  or  calculate  by  Tri- 
gonometry, AG.  Let  Qbe 
the  quadrilateral  ABCD,  and, 
as  before,  a'  =  BCG.     Then 

GE  =  AGy|^J^|;  GL^AG^/j"^  +  V|;  ' 
(  a'  +  Q  )  (  a'  +  Q  ) 

GN  =  AG^I  j      T  [ ;  -^c. 

i   a   +Q    ) 
If  the  quadrilateral  be  given  by  Bearmgs,  part  off,  by  Art.  (504), 

1  '^ 

-  .  ABCD,  then  part  off  -  .  ABCD  ;  &c. ;  so  in  any  similar  case. 

F  P 


«58 


LAYING  on  AXD  DIVIDLXG  TP  LAND,      [part  xi 


(549)  By  lines  perpendicular  to  a  side. 

quadrilateral  which  is  to  be  divided,  by 
a  line  perpendicular  to  AD,  into  two 
parts  having  a  ratio  =  m  :  n.  By  hypo- 
thesis, ABEF  =  -^^  .  ABCD. 


Let  ABCD  be  a 

Fig.  387. 


Ji 


m  A-  n  ^ 

Taking  away  the  triangle  ABG,  the  ^  ^ 
remainder,  GBEF,  will  be  to  the  rest  of  the  figure  in  a  known 
ratio,  and  the  position  of  EF,  parallel  to  BG,  will  be  found  as  in 
the  last  article. 


(550)  By  lines  running  in  any  given  direction!    To  diAade 

a  quadrilateral  ABCD  into  two  parts  : :  m  :  n,  part  off  from  it  an 

m 


area 


m  +  n 


ABCD,  by  the  methods  of  Arts.  (509)  or  (510), 


if  the  area  parted  off  is  to  be  a  triangle,  or  Arts.  (511)  or  (512), 
if  the  area  parted  off  is  to  be  a  quadrilateral. 


(551)   By  lines  starting  from  an  angle 

divided,  by  the  line  CE,  into  two 
parts  having  the  ratio  m  :  n. 
Since  the  area  of  the  triangle 


CDE  = 


ABCD,  DE  will 


m  +  n 

be  obtained  by  dividing  this  area 
by  half  of  the  altitude  CF. 


(552)  By  lines  starting  from  points  in  a  side.      Let  it    be 

required  to  divide  ABCD  into  two  Fig.  389. 

parts  : :  m  :  n,  by  a  line  starting  from 
the   point   E.     The   area  ABFE  is 

known,  (being  =  — ^? —  .  ABCD)  as 
m  +  n 

also  ABE;  AB,  BE,   and  EA  be-    a 

ing  given  on  the  ground.     BEF  will  then  be  known  =  ABFE  -  - 

ABE.     Then  GF  =  j^i  and  the  point  F  is  obtained  by  running 
a  parallel  to  BE,  at  a  perpendicular  distance  from  it  =  GF. 


3HAP.  III.] 


Dividing  up  Laud. 


359 


Fis.  391. 


To  divide  a  quadrilateral,  ABCD,  Fig.  39o. 

graphically,  into  two  equivalent  parts 
by  a  line  from  a  point,  E,  on  a 
side,  proceed  thus.  Draw  the  diago- 
nal CA,  and  from  B  draw  a  parallel 
to  it,  meeting  DA  prolonged  in  F. 
Mark  the  middle  point,  G,  of  FD.    F^  a     g      h 

Join  GE.  From  C  draw  a  parallel  to  EG,  meeting  DA  in  H.  EH 
is  the  required  line.  The  quadrilateral  could  also  be  divided  in 
any  ratio  =  m  :  n,  by  dividing  FD  in  that  ratio. 

If  the  quadrilateral  be  given  by  Bearings,  proceed  to  part  off 
the  desired  area,  as  in  Art.  (515)  or  (516). 

(553)  Let  it  be  required  to  divide  a  quadrilateral,  ABCD,  into 
three  equivalent  parts. 
From  any  angle,  as  C, 
draw  CE,  parallel  to  DA. 
Divide  AD  and  EC,  each 
into  three  equal  parts,  at 

F,  F,  and  G,  G'.     Draw 
BF,  BF'.     From  G  draw 

GH,  parallel  to  FB,  and       

from  G'  draw  G'H',  pa-    ^  F  *" 

rallel  to  F'B.     FH  and  F'H'  are  the  required  lines  of  division. 

Let  it  be  required  to  make 
the  above  division  by  lines 
starting  from  '  two  given 
■points,  P  and  Q.  Reduce 
the  quadrilateral  to  an  equi- 
valent triangle  CBE,  as  in 
Art.  (87).  Divide  EB  into 
three  equal  parts  at  F  and     /sl Z 

G.  Join  CQ,  and,  from  G, 
draw  GK  parallel  to  it.     Jom  CP,  and  from  F  draw  FL  parallel 
to  it.-     Join  PL  and  QK,  and  they  will  be  the  division  Unes  required. 

(554)  By  liues  passing  through  a  point  within  the  figure. 

Proceed  to  part  off  the  desired  area  as  in  Arts.  (519),  (520),  or 
(521),  according  to  the  circumstances  of  the  case. 


FiR.  392. 


S60 


LiriNG  OUT  AND  DIVIDING  UP  LAND.       [part  xi 


DIVISION    OF   POLYGONS. 

(555)  By  lines  running  in  any  direction.  Let  ABCDEFG  be 
a  given  polygon,  and  BH  the  di- 
rection parallel  to  -vvhicli  is  to  be 
drawn  a  line  PQ,  dividing  the 
polygon  into  two  parts  in  any  de- 
sired ratio  =  m  :  n.     The   area 


PCDEQ  = 


ABCDEFG. 


m  -\-  n 
Taking  it  from  the  area  BCDEH, 
the  remainder  will  be  the  area 
BPQH.  The  quadrilateral 
BCEH,  CE  being  supposed  to  be  drawn,  can  then  be  divided  by 
the  method  of  Art.  (548),  into  two  parts,  BPQH  and  PQEC, 
having  to  each  other  a  known  relation. 

If  DK  were  the  given  direction,  at  right  angles  to  the  former, 
the  position  of  a  dividing  Ime  RS  could  be  similarly  obtained. 


(556)  By  lines  starting  from  an  angle. 

Fig.  394. 
Z 


Produce  one  side,  AB 


X  A  W  B  P 

of  the  given  polygon,  both  ways,  and  reduce  the  polygon  to  a  single 
equivalent  triangle,  XYZ,  by  the  method  of  Art.  (82).  Then 
divide  the  base,  XY,  in  the  required  ratio,  as  at  W,  and  draw 
ZW,  which  will  be  the  division  line  desired.  In  this  figure  the 
polygon  is  divided  into  two  equivalent  parts. 


CHa.'.  III.] 


DividiD!^  lip  Laud. 


3GI 


If  the  division  line  should  pass  outside  of  the  polygon,  as  does 
ZP,  through  P  draw  a  parallel  to  BZ,  meeting  the  adjacent  side 
of  the  polygon  in  Q,  and  ZQ  will  be  the  division  line  desired. 


(557)  By  lines  starting  from  a  point  on  a  side* 

(517)  and  (518)  in  the  preceding  chapter. 


See  Articles 


(558)  By  lines  passing  through  a  point  within  the  Gguret 

Part  off,  as  in  Arts.  (519)  or  (522)  in  the  preceding  chapter, 
if  a  straight  line  be  required  ;  or  by  guess  Imes  and  the  addition 
of  triangles,  as  in  Art.  (538)  of  this  chapter,  if  the  lines  have 
merely  to  start  from  the  point,  such  as  a  spring  or  well. 

(559)  Other  problems.  The  following  is  from  Gummere's  Sur- 
veying. Question.  A  tract  of  land  is  Fig.  39.5. 
bounded  thus :  N.  35^^  E.,  23.00  ;  N. 
751^  E.,  30.50  ;  S.  3^^  E.,  46.49  ;  N. 
QQ^"^  W.,  49.64.  It  is  to  be  divided  into 
four  equivalent  parts  by  two  straight  hnes, 
one  of  which  is  to  run  parallel  to  the  third 
side  ;  required  the  distance  of  the  parallel 
division  line  from  the  first  corner,  mea- 
sured on  the  fourth  side  ;  also  the  Bearmg 
of  the  other  division  line,  and  its  distance  from  the  same  corner 
measured  on  the  first  side.  Ans.  Distance  of  the  parallel  divi- 
sion line  from  the  first  corner,  32.50 ;  the  Bearing  of  the  other, 
S.  88^  22'  E. ;  and  its  distance  from  the  same  corner  5.99. 

The  scale  of  the  figure  is  40  chains  to  1  inch  =  1 :  31680. 


An  indefinite  number  of  problems  on  this  subject  might  be  pro 
posed,  but  they  would  be  matters  of  curiosity  rather  than  of  utility, 
and  exercises  in  Geometry  and  Trigonometry  rather  than  in  Sur- 
veying ;  and  the  youngest  student  will  find  his  life  too  short  for 
even  the  hastiest  survey  of  merely  the  most  fruitful  parts  of  the 
boundless  field  of  Mathematics. 


862 


U.  S.  PUBLIC  LANDS. 

Fie.  396. 


PART  XII. 


^BlThLY  ajMtLYH.SI. 


PART  XII. 

THE    PUBLIC   LANDS 
OF  THE    UNITED   STATES." 

(560)  Geueral  systenii  The  Public  Lands  of  the  United  States 
of  America  are  generally  divided  and  laid  out  into  squares,  the 
Bides  of  which  run  truly  North  and  South,  or  East  and  West. 

This  is  effected  by  means  of  Meridian  lines  and  Parallels  of  Lati- 
tude, established  six  miles  apart.  The  principal  meridians  and  base 
lines  are  established  astronomically,  and  the  intermediate  ones  are 
run  with  chain  and  compass.  The  squares  thus  formed  are  called 
Townships.  They  contain  36  square  miles,  or  23040  acres,  "  as 
nearly  as  may  be."  The  map  on  the  opposite  page  represents  a 
portion  of  the  Territory  of  Oregon  thus  laid  out.  The  scale  is  10 
miles  to  1  inch  =  1 :  633600.  On  it  will  be  seen  the  "Willamette 
Mei'idian,"  running  truly  North  and  South,  and  a  "  Base  line," 
which  is  a  "  Parallel  of  Latitude,"  running  truly  East  and  West. 
Parallel  to  these,  and  six  miles  from  them,  are  other  lines,  forming 
Townships.  All  the  Townships,  situated  North  or  South  of  each 
i/Cher,  form  a  Range.  The  Ranges  are  named  by  their  number 
East  or  West  of  the  principal  Meridian.  In  the  figure  are  seen 
three  Ranges  East  and  West  of  the  Willamette  Meridian.  They  are 
noted  as  R.  I.  E.,  R.  I.  W.,  &c.  The  Townships  in  each  Range 
are  named  by  their  number  North  or  South  of  the  Base  line.     In 

*  The  substance  of  this  Part  is  mainly  taken  fi-oni  "  Instructions  to  the  Surveyor 
General  of  Oregon,  being  a  Manual  for  Field  Operations,"  prepared,  in  March, 
1851,  by  John  M.  Moore,  "  Principal  Clerk  of  Surveys,"  by  direction  of  Hon. 
J.  Butterfiekl,  "  Commissioner  of  the  General  Land  Office,"  and  communicated  to 
the  author  by  Hon.  John  Wilson,  the  present  Commissioner.  The  aim  of  the 
"  Instructions"  is  stated  to  be  "  simplicity,  uniformity  and  permanency."  They 
»eem  admirably  adapted  for  these  objects,  and  the  lastinsr  importance  of  tlie  subject 
in  this  country  has  led  the  author  to  reproduce  about  half  of  them  in  this  place. 
They  were  subsequently  directed  to  be  adopted  for  tlie  Surveying  service  in 
Minnesota  and  California. 


364 


U.  S.  PUBLIC  LAINDS. 


[part  XII. 


N 


W 


6 
7 

18 
19 
30 
31 

5 

8 

I? 
20 
29 
32 

4 
9 

16 
21 

28 
38 

3 

10 
•15 

22 
27 
34 

2 

n 
n 

23 
26 
35 

1 
l2 

13 
24 
25 
36 

E 


S 


the  figure  along  the  principal  Meridian  are  seen  four  North 
and  five  South  of  the  Base  hne.  Thej  are  noted  as  T.  1  N., 
T.  2  N.,  T.  1  S.,  &c.* 

Each  Township  is  divided  into  36  Sec- 
tions, each  1  mile  square,  and  therefore 
containing,  "  as  nearly  as  may  be,"  640 
acres.  The  sections  in  each  Township  are 
numbered,  as  in  the  margin,  from  1  to  36, 
beginning  at  the  North-east  angle  of  the 
Township,  and  going  West  from  1  to  6, 
then  East  from  7  to  12,  and  so  on  alter- 
nately to  Section  36,  which  will  be  in  the  South-east  angle  of  the 
Township.  The  Sections  are  sub-divided  into  Quarter-sections, 
half-a-mile  square,  and  containing  160  acres,  and  sometimes  into  half- 
quarter-sections  of  80  acres,  and  quarter-quarter-sections  of  40  acres. 

By  this  beautiful  system,  the  smallest  subdivision  of  land  can  be 
at  once  designated ;  such  as  the  North-east  quarter  of  Section  31, 
in  Township  two  South,  in  range  two  East  of  Willamette  Meridian. 

(561)  Difficnlty.  "  The  law  requires  that  the  lines  of  the 
public  surveys  shall  be  governed  by  the  true  meridian,  and  that 
the  townships  shall  be  six  miles  square,  —  two  things  involving  in 
connection  a  mathematical  impossibility — for,  strictly  to  conform 
to  the  meridian,  necessarily  throws  the  township  out  of  square,  by 
reason  of  the  convergency  of  meridians ;  hence,  adhering  to  the 
true  meridian  renders  it  necessary  to  depart  from  the  strict  requu'e- 
ments  of  law  as  respects  the  precise  area  of  townships,  and  the 
subdivisional  parts  thereof,  the  township  assuming  something  of  a 
trapezoidal  form,  which  inequality  developes  itself,  more  and  more 
as  such,  the  higher  the  latitude  of  the  surveys.  In  view  of  these 
circumstances,  the  law  provides  that  the  sections  of  a  mile,  square 
shall  contain  the  quantity  of  640  acres,  as  nearly  as  may  he;  and, 
moreover,  provides  that  '  In  all  cases  where  the  exterior  fines  of 
the  townships,  thus  to  be  subdivided  into  sections  or  half-sections, 
shall  exceed,  or  shall  not  extend,  six  miles,  the  excess  or  deficiency 


•  The  marks  O,  -|-  and  A  ,  merely  refer  to  the  dates  of  the  surveys.     Tlicy  are 
•ometimes  used  to  point  out  lands  offered  for  sale,  or  reserved.  &c. 


PA^RT  xii]  Difficulty.  S65 

Bhall  be  specially  noted,  and  added  to  or  deducted  from  the  western 
or  northern  ranges  of  sections  or  half-sections  in  such  township, 
according  as  the  error  may  be  in  running  the  lines  from  east  to 
west,  or  from  south  to  north.' " 

"  In  order  to  throw  the  excesses  or  deficiencies,  as  the  case  may 
be,  on  the  north  and  on  the  west  sides  of  a  township,  according  to 
law,  it  is  necessary  to  sui'vey  the  sectmt  lines  from  south  to  north 
on  a  true  meridian,  leaving  the  result  in  the  northern  line  of  the 
township  to  be  governed  by  the  convexity  of  the  earth  and  the 
convergency  of  meridians." 

Thus,  suppose  the  land  to  be  surveyed  lies  between  46^  and  47^ 
of  North  Latitude.  The  length  of  a  degree  of  Longitude  in  Lat. 
46^  N.  is  taken  as  48.0705  statute  miles,  and  in  Lat.  47°  N.  as 
47.1944.  The  difference,  or  convergency  per  square  degree  = 
0.8761  =  70.08  chains.  The  convergency  per  Range  (8  per 
degree  of  Longitude)  equals  one-eighth  of  this,  or  8.76  chains ; 
and  per  Township  (11 1  per  degree  of  Latitude)  equals  the  above 
divided  by  11|,  i.  e.  0.76  chain.  We  therefore  know  that  the 
width  of  the  Townships  along  their  Northern  line  is  76  hnks  lesa 
than  on  their  Southern  line.  The  townships  North  of  the  base  Ime 
therefore  become  narrower  and  narrower  than  the  six  mile  width 
with  which  they  start,  by  that  amount ;  and  those  South  of  it  as 
much  wider  than  six  miles. 

"  Standard  Parallels  (usually  called  correction  lines'),  are 
established  at  stated  intervals  (24  or  30  miles)  to  provide  for  or 
counteract  the  error  that  otherwise  would  result  from  the  conver- 
gency of  meridians ;  and,  because  the  public  surveys  have  to  be 
governed  by  the  true  meridian,  such  lines  serve  also  to  arrest  error 
arising  from  inaccuracies  in  measurements.  Such  lines,  when  Ijnng 
north  of  the  principal  base,  themselves  constitute  a  base  to  the  sur- 
veys on  the  north  of  them ;  and  where  lying  south  of  the  prin- 
cipal base,  they  constitute  the  base  for  the  surveys  south  of  them." 

The  convergency  or  divergency  above  noticed  is  taken  up  on 
these  Correction  hues,  from  which  the  townships  start  again  with 
their  proper  widths.  On  these  therefore  there  are  found  Double 
Corners,  both  for  Townships  and  Sections,  one  set  being  the 
Closing  Corners  of  the  surveys  ending  there,  and  the  other  set 
being  the  Standard  Corners  for  the  surveys  starting  there. 


366  U.  S.  PUBLIC  LAi\DS.  [partxii. 

(562)  Running  Township  lines.  "  The  principal  meridian,  tha 
base  line,  and  the  standard  parallels  having  been  first  astronomi- 
cally run,  measured,  and  marked,  according  to  instructions,  on  true 
meridians,  and  true  parallels  of  latitude,  the  process  of  running, 
measuring,  and  marking  the  exterior  lines  of  townships  will  be  as 
follows. 

Townships  situated  NORTH  of  the  base  line,  and  west  of  the 
princijjal  meridian*  Commence  at  No.  1,  being  the  southwest 
corner  of  T.  1  N.- — R.  1  W.,  as  established  on  the  base  line; 
thence  run  north,  on  a  true  meridian  line,  four  hundred  and  eighty 
chains,  establishing  the  mile  and  half-mile  corners  thereon,  as  per 
instructions,  to  No.  2,  (the  northwest  corner  of  the  same  townsliip), 
whereat  establish  the  corner  of  Tps.  1  and  2  N. —  Rs.  1  and  2 
W. ;  thence  east,  on  a  random  or  trial  line,  setting  temporary  mik 
and  half-mile  stakes  to  No.  3,  (the  northeast  corner  of  the  same 
township),  where  measure  and  note  the  distance  at  which  the 
line  intersects  the  eastern  boundary,  north  or  south  of  the  true 
or  established  corner.  Rim  and  measure  westivard,  on  the  true 
line,  (taking  care  to  note  all  the  land  and  water  crossings,  &c.,  as 
per  instructions),  to  No.  4,  which  is  identical  with  No.  2,  establish- 
ing the  mile  and  half-mile  permanent  corners  on  said  line,  the 
last  half-mile  of  which  will  fall  short  of  being  forty  chains,  by  about 
the  amount  of  the  calculated  convergency  per  township,  76  linka 
in  the  case  above  supposed.  Should  it  ever  happen,  however,  that 
such  random  line  materially  falls  short,  or  overruns  in  length,  or 
intersects  the  eastern  boundary  of  the  township  at  any  considerable 
distance  from  the  true  corner  thereon,  (either  of  which  would  indi- 
cate an  important  error  in  the  surveying),  the  lines  must  be  retraced, 
even  if  found  necessary  to  remeasure  the  meridional  boundaries  of 
the  township  (especially  the  western  boundary),  so  as  to  discover 
and  correct  the  error ;  in  doing  which,  the  true  corners  must  be 
estaJ)lished  and  marked,  and  the  false  ones  destroyed  and  oblite- 
rated, to  prevent  confusion  in  future ;  and  all  the  facts  must  be 
distinctly  set  forth  in  the  notes.  Thence  proceed  in  a  similar 
manner  north,  from  No.  4  to  No.  5,  (the  N.  W.  corner  of  T.  2  N. 
— R.  1  W.),  east  from  No.  5  to  No.  6,  (the  N.  E.  corner  of  the 
same  township),  west  from  No.  6  to  No.  7,  (the  same  as  No.  5), 
north  from  No.  7  to  No.  8,  (the  N.  W.  corner  of  T.  3  N.,  R.  1  W.), 
east  from  No.  8  to  Na.  9,  (the  N.  E.  corner  of  same  township),  and 
thence  west  to  No.  10,  (the  same  as  No.  8),  or  the  southwest  corner 
T.  4  N. — R.  1  W.  Thence  north,  still  on  a  true  meridian  line, 
establishing  the  mile  and  half-mile  corners,  until  reaching  the 
standard  par  ^llel  or  correction  line,  (which  is  here  four  town- 

»  The  Surveyor  should  prepare  a  diagram  of  the  townships,  with  the  number* 
here  referred  t(».  in  their  proper  places,  as  here  indicated 


PARTxii  j  Runuing  Township  Lines.  307 

ships  north  of  the  base  line);  throwing  the  excess  over,  or  deficiency 
under,  four  hundred  and  eighty  chains,  on  the  last  half-mile, 
according  to  law,  and  at  the  intersection  establishing  the  "  closing 
CORNER,"  the  distance  of  which  from  the  standard  corner  must  be 
measured  and  noted  as  required  bj  the  instructions.  But  should 
it  ever  so  happen  that  some  impassable  barrier  will  have  prevented 
>or  delayed  the  extension  of  the  standard  parallel  along  and  above 
the  field  of  present  survey,  then  the  surveyor  will  plant,  in  place, 
the  corner  for  the  township,  subject  to  correction  thereafter,  should 
such  parallel  be  extended. 

Toivnshijys  situated  north  of  the  base  line,  and  east  of  the 
principal  meridian.  Commence  at  No.  1,  being  the  southeast 
corner  of  T.  1  N. — R.  1  E.,  and  proceed  as  with  townships  situa- 
ted "  north  and  west,"  except  that  the  random  or  trial  hues  will  be 
run  and  measured  zvest,  and  the  true  lines,  east,  throwing  the 
excess  over  or  deficiency  under  four  hundred  and  eighty  chains  on 
the  west  end  of  the  line,  as  required  by  law ;  wherefore,  the  sur- 
veyor will  commence  his  measurement  with  the  length  of  the  defi- 
cient or  excessive  half-section  boundary  on  the  west  of  the  town- 
sliip,  and  thus  the  remaining  measurements  will  all  be  even  miles 
and  half-miles. 

Toumships  situated  SOUTH  of  the  base  line,  and  west  of  the 
principal  meridian.  Commence  at  No.  1,  the  northivest  corner 
of  township  1  S.,  range  1  W*,  and  proceed  due  south  in  running 
and  measuring  line,  establishing  and  marking  the  mile,  half-mile, 
and  township  corners  thereon,  precisely  in  the  method  prescribed 
for  running  north  and  west,  with  the  exception  that,  in  order 
to  thi'ow  the  excess  or  deficiency  (over  or  under  four  hundred  and 
eighty  chains)  of  the  western  boundaries  of  such  of  those  townships 
as  close  on  the  standard  parallel  on  the  south,  upon  the  most 
nortliern  half-mile  of  the  townships,  according  to  law,  the  proceed- 
ing will  be  as  follows. 

The  western  (meridional)  boundary  line  of  every  township, 
closing  on  the  standard  jyarallel,  (being  every  fifth  one  in  this 
case),  will  be  carefully  run  soidh,  on  a  true  meridian,  until  it  inter- 
sects the  standard,  planting  stakes  and  making  distinctive  marks 
on  line  trees,  in  sufficient  number  to  serve  as  giddes  in  afterwards 
retracing  the  line  7i07-th  with  ease  and  certainty.  At  the  point  of 
the  line's  intersection  of  the  standard,  the  survej'or  will  establish 
the  "  closing''  (southwest)  comer  of  the  township,  noting  in  his 
field-book  its  distance  and  direction  from  the  "  standard  corner." 
Then  starting  from  such  "  closing  corner,"  he  will  proceed  north 
on  the  line  identified  by  the  guide  stakes  and  marks,  measuring 
Buch  fine,  and  estabhshing  thereon  the  mile  and  half-mile  stations, 
and  noting,  as  he  goes,  all  the  land  and  water  crossings,  kc. 


368 


r.  S.  PUBLIC  LANDS. 


[part  xh. 


Townships  situated  south  of  the  base  line,  and  east  of  the 
principal  meridian.  Commence  at  No.  1,  at  the  northeast  corner 
of  township  1  S.,  range  1  E.,  and  proceed  precisely  as  with  the 
townships  situated  "south  and  west,"  except  that  the  random 
lines  will  be  run  and  measured  west,  and  the  true  Imes  east ;  the 
deficiency  or  excess  of  the  measurements  being,  as  in  all  other 
cases,  thrown  upon  the  most  western  half-mile  of  Une." 

(563)  Runuing  Section  Hues.  The  interior  or  sectional  lines 
of  all  townships,  however  situated  in  reference  to  the  Base  and 
Meridian  hues,  are  laid  off  and  surveyed  as  below. 

31  32  33  34  35  36 


12 


13 


24 


25 


36 


97 

71 

53 

35 

17 

6 

5 

4 

3 

2 

1 

99 

98 

96 

72 

70 

54 

52 

36 

34 

18 

16 

lUO 

94 

95 

(iS 

69 

50 

51 

32 

33 

14 

15 

7 

8 

9 

10 

11 

12 

1)2 

93 

91 

ca 

49 

31 

13 

89 

90 

()5 

66 

47 

48 

29 

30 

11 

12 

18 

17 

16 

15 

14 

13 

87 

m 

CA 

46 

28 

10 

88 

84 

85 

62 

63 

44 

45 

26 

27 

8 

9 

19 

20 

21 

22 

23 

24 

82 

81 

61 

43 

25 

7 

8:J 

79 

80 

59 

60 

41 

42 

23 

24 

5 

6 

30 

29 

'28 

27 

26 

25 

77 

7(1 

58 

40 

22 

4 

78 

74 

75 

56 

57 

38 

39 

20 

2] 

o 

3 

31 

73 

32 

55 

33 

37 

34 

19 

35 

1 

36 

18 


19 


30 


31 


In  the^  above  Diagram,  the  squares  and  large  figures  repre- 
sent sections,  and  the  small  figures  at  their  corners  are  those 
referred  to  in  the  following  directions. 

"  Commence  at  No.  1,  (see  small  figures  on  diagram),  the  cor- 
ner established  on  the  township  boundary  for  sections  1, 2,  35,  and 


PART  XII. ]  Runuiug  Section  Lines.  3G9 

86 ;  thence  run  north  on  a  true  meridian ;  at  40  chains  settin" 
the  half-mile  or  quarter-section  post,  and  at  80  chains  (No.  2) 
establishing  and  marking  the  corner  of  sections  25,  26, 35,  and  36. 
Thence  east,  on  a  random  line,  to  No.  3,  setting  the  temporary 
quarter-section  post  at  40  chains,  noting  the  measurement  to  No.  3, 
and  the  measured  distance  of  the  random's  intersection  7iorth  or 
south  of  the  true  or  estabhshed  corner  of  sections  25,  36,  30,  and 
31,  on  the  township  boundary.  Thence  correct,  west,  on  the  trut 
line  to  No.  4,  setting  the  quarter-section  post  on  this  line  exactly 
at  the  equidistant  point,  now  known,  between  the  section  corners 
indicated  by  the  small  figures  Nos.  3  and  4.  Proceed,  in  hke 
manner,  from  No.  4  to  No.  5,  5  to  6,  6  to  7,  and  so  on  to  No.  16, 
the  corner  to  sections  1,  2,  11,  and  12.  Thence  7iorth,  on  a  ran- 
dom line,  to  No.  17,  setting  a  temporary  quarter-section  post  at  40 
chains,  noting  the  length  of  the  whole  luae,  and  the  measured  dis- 
tance of  the  random's  intersection  east  or  icest  of  the  true  corner 
of  sections  1,  2,  35,  and  36,  established  on  the  township  boundary , 
thence  southivardly  from  the  latter,  on  a  true  line,  noting  the 
course  and  distance  to  No.  18,  the  established  corner  to  sections 
1,  2,  11,  and  12,  taking  care  to  establish  the  quarter-section 
corner  on  the  true  line,  at  the  distance  of  40  chains  from  said  sec- 
tion corner,  so  as  to  throw  the  excess  or  deficiency  on  the  northern 
half-mile,  according  to  law.  Proceed  in  like  maimer  through  all 
the  intervening  tiers  of  sections  to  No.  73,  the  corner  to  sections 
31,  32,  5,  and  6  ;  thence  north,  on  a  true  meridian  hne,  to  No, 
74,  establishing  the  quarter-section  corner  at  40  chains,  and  at  80 
chains  the  corner  to  sections  29,  30,  31,  and  32 ;  thence  east,  on 
a  random  line  to  No.  75,  setting  a  temporary  quarter-section  post 
at  40  chains,  noting  the  measurement  to  No.  75,  and  the  distance 
of  the  random's  intersection  north  or  south  of  the  established  corner 
of  sections  28,  29,  32,  and  33  ;  thence  loest  from  said  corner,  on 
the  true  line,  setting  the  quarter-section  post  at  the  equidistant 
point,  to  No.  76,  which  is  identical  with  74 ;  thence  west,  on  a 
random  line,  to  No.  77,  setting  a  temporary  quarter-section  post 
at  40  chains,  noting  the  measarement  to  No,  77,  and  the  distance 
of  the  random's  intersection  with  the  western  boundary,  noi'th  oi 
south  of  the  established  corner  of  sections  25,  36,  30,  and  31 ;  and 
from  No.  77,  correct,  eastward,  on  the  true  line,  giving  its  course, 
but  establishing  the  quarter-section  post,  on  this  line,  so  as  to 
retain  the  distance  of  40  chams  from  the  corner  of  sections  29, 
30,  31,  and  32 ;  thereby  throwing  the  excess  or  deficiency  of  me.v 
Burement  on  the  most  western  half-mile.  Proceed  north,  in  a  simi- 
lar manner,  from  No.  78  to  79,  79  to  80,  80  to  81,  and  so  on  to 
96,  the  south-east  comer  of  section  6,  where  having  estabhshed  tho 
comer  for  sections,  5,  6,  7,  and  8,  run  thence,  successively,  3D 

24 


S70  U.  S.  PUBLIC  LANDS.  [part  xii 

random  line  east  to  95,  north  to  97,  and  west  to  99 ;  and  bj 
reverse  coui'ses  correct  on  true  lines  bach  to  said  south-east  comer 
of  section  6,  establishing  the  quarter-section  corners,  and  noting 
the  courses,  distances,  &c.,  as  before  described. 

In  townships  contiguous  to  standard  parallels,  the  above  method 
will  be  varied  as  follows.  In  every  township  south  of  the  princi- 
pal base  line,  which  closes  on  a  standard  parallel,  the  surveyor  will 
begin  at  the  south-east  corner  of  the  township,  and  measure  west  on 
the  standard,  establishing  thereon  the  mile  and  half-mile  corners, 
and  noting  their  distances  from  the  pre-established  corners.  He 
then  will  proceed  to  subdivide,  as  directed  under  the  above  head. 

In  the  townships  north  of  the  principal  base  line,  which  close 
on  the  standard  parallel,  the  sectional  lines  must  be  closed  on  the 
standard  by  true  meridians,  instead  of  by  course  lines,  as  directed 
under  the  above  head  for  townships  otherwise  situated ;  and  the 
connexions  of  the  closing  corners  with  the  pre-established  standard 
corners  are  to  be  ascertained  and  noted.  Such  procedure  does 
away  with  any  necessity  for  running  the  randoms.  But  in  case 
he  is  unable  to  close  the  lines  on  account  of  the  standard  not  hav- 
ing been  run,  from  some  inevitable  necessity,  as  heretofore  men- 
tiond,  he  will  plant  a  temporary  stake,  or  mound,  at  the  end  of  the 
sixth  mile,  thus  leaving  the  lines  and  their  connexions  to  be  finished, 
and  the  ijermanent  corners  to  be  planted,  at  such  time  as  the 
standard  shall  be  extended." 

(564)  Exceptional  methOdSi  Departures  from  the  general  sys- 
tem of  subdividing  public  lands  have  been  authorized  by  law  in 
certain  cases,  particularly  on  water-fronts. 

Thus,  an  act  of  Congress,  March  3,  1811,  authorized  the  sur- 
veyors of  Lousiana,  "  in  surveying  and  dividing  such  of  the  pub- 
lic lands  in  the  said  territory,  which  are  or  may  be  authorized 
to  be  surveyed  and  divided,  as  are  adjacent  to  any  river,  lake, 
creek,  bayou,  or  water  course,  to  lay  out  the  same  into  tracts,  as 
far  as  practicable,  of  fifty-eight  poles  in  front,  and  four  hundred 
and  sixty-five  poles  in  depth,  of  such  shape,  and  bounded  by  such 
lines,  as  the  nature  of  the  country  will  render  practicable  and  most 
convenient."  Another  act,  of  May  24,  1824,  authorizes  lands 
similarly  situated  "to  be  surveyed  in  tracts  of  two  acres  in  width, 
fronting  on  any  river,  bayou,  lake,  or  water  course,  and  running 
back  the  depth  of  forty  acres ;  which  tracts  of  land,  so  surveyed, 
Bhall  be  oflFered  for  sale  entire,  instead  of  in  half-quarter-sections." 

The  "Instructions"  from  which  we  have  quoted  say,  "  In  those 
localities  where  it  would  best  subserve  the  inter^ts  of  the  people 
to  have  fronts  on  the  navigable  streams,  and  to  run  back  into  the 


PART  XII. ]  Excoptioual  3Iethods.  371 

uplands  for  quantity  and  timber,  the  principles  of  the  act  of  May 
24th,  1824,  may  be  adopted,  and  you  are  authorized  to  enlarge  the 
quantity,  so  as  to  embrace  four  acres  front  by  forty  in  depth,  form 
ing  tracts  of  one  hundred  and  sixty  acres.  But  in  so  doing  it  ia 
designed  only  to  survey  the  lines  between  every  four  lots,  (or  640 
acres),  but  to  estabhsh  the  boundary  posts,  or  mounds,  in  front 
and  i?i  rear,  at  the  distances  requisite  to  secure  the  quantity  of  160 
acres  to  each  lot,  either  rectangularly,  when  practicable,  or  at 
obUque  angles,  when  otherwise.  The  angle  is  not  important,  so 
that  the  principle  be  maintained,  as  far  as  practicable,  of  making 
the  work  to  square  in  the  rear  with  the  regular  sectioning. 

The  numbering  of  all  anomalous  lots  -^-ill  commence  with  No.  37, 
to  avoid  the  possibility  of  conflict  with  the  numbering  of  the  regular 
sections." 

The  act  of  Sept.  27,  1850,  authorized  the  Department,  should 
it  deem  expedient,  to  cause  the  Oregon  surveys  to  be  executed 
according  to  the  principles  of  what  is  called  the  "Geodetic  Method." 
The  complete  adoption  of  this  has  not  been  thought  to  be 
expedient ;  but  "  it  was  deemed  useful  to  institute  on  the  principal 
base  and  meridian  Hues  of  the  public  surveys  in  Oregon,  ordered 
to  be  established  by  the  act  referred  to,  a  system  of  triarigulations 
from  the  recognized  legal  stations,  to  all  prominent  objects  within 
the  range  of  the  theodolite  ;  by  means  of  which  the  relative  dis- 
tances of  such  objects,  in  respect  to  those  main  lines,  and  also  to 
each  other,  might  be  observed,  calculated,  and  protracted,  with 
the  view  of  contributing  to  the  knowledge  of  the  topography  of  the 
country  in  advance  of  the  progressuig  Unear  surveys,  and  to  obtain 
the  elements  for  estimating  areas  of  valleys  intervening  between 
the  spurs  of  the  mountains." 

"  Meandering"  is  a  name  given  to  the  usual  mode  of  surveying 
with  the  compass,  particularly  as  applied  to  navigable  streams. 
The  "  Instructions"  for  this  are,  in  part,  as  follows. 

"  Both  banks  of  navigable  rivers  are  to  be  meandered  bj  taking 
the  courses  and  distances  of  their  sinuosities,  and  the  same  are  to 
be  entered  in  the  ^Meander  field-book.'  At  those  pomts  where 
either  the  township  or  section  lines  intersect  the  banks  of  a  navi- 
gable stream,  posts,  or,  where  necessary,  mounds  of  earth  or  stone, 
(as  noted  in  Art.  (566,))  are  to  be  estabhshed  at  the  time  of 
running  these  lines.  These  are  called  "  meander  corners ;"  and 
in  meandering  you  are  to  commence  at  one  of  those  corners  on  the 
jownship  hne,  coursing  the  banks,  and  measuring  the  distance  of 
aach  course  from  your  commencing  corner  to  the  next  '  meandei 


372  r.  S.  PrBLIC  LA\BS.  [part  in 

corner,'  u|)on  the  same  or  another  boundary  of  the  same  township ; 
carefully  noting  your  intersection  with  all  intermediate  meander 
corners.  By  the  same  method  you  are  to  meander  the  opposite 
bank  of  the  same  river. 

The  crossing  distance  between  the  meander  corners,  on  same 
line,  is  to  be  ascertained  by  triangulation,  in  order  that  the  river 
may  be  protracted  with  entire  accuracy.  The  particulars  to  be 
given  in  the  field-notes. 

The  courses  and  distances  on  meandered  navigable  streams, 
govern  the  calculations  wherefrom  are  ascertained  the  true  areas 
of  the  tracts  of  land  (sections,  quarter  sections,  kc.^  known  to  the 
law  aj&  fractional,  and  bounding  on  such  streams." 

You  are  also  to  meander,  in  manner  aforesaid,  all  lakes  and 
deep  ponds  of  the  area  of  twentj^-five  acres  and  upwards ;  also 
navigable  bayous. 

The  precise  relative  position  of  islands,  in  a  township  made 
fractional  by  the  river  in  which  the  same  are  situated,  is  to  be 
determined  trigonometrically.  Sighting  to  a  flag  or  other  fixed 
object  on  the  island,  from  a  special  and  carefully  measured  base 
line,  connected  with  the  surveyed  lines,  on  or  near  the  river  bank, 
you  are  to  form  connexion  between  the  meander  corners  on  the 
river  to  points  corresponding  thereto,  in  direct  line,  on  the  bank 
of  the  island,  and  there  establish  the  proper  meander  corners,  and 
calculate  the  distance  across." 

(.565)  Marking  Lines.  "AH  lines  on  which  are  to  be  estab- 
lished the  legal  corner  boundaries,  are  to  be  marked  after  this 
method,  viz :  Those  trees  which  may  intercept  your  line,  must 
have  two  chops  or  notches  cut  on  each  side  of  them  without  any 
other  marks  whatever.  These  are  called  '  sight  trees,^  or  '  line  trees.' 

A  sufficient  number  of  other  trees  standing  nearest  to  your  line, 
on  either  side  of  it,  are  to  be  blazed  on  two  sides,  diagonallj*  or 
quartering  towards  the  line,  in  order  to  render  the  line  conspicu- 
ous, and  readily  to  be  traced,  the  blazes  to  be  opposite  each  other, 
coinciding  in  direction  with  the  hne  where  the  trees  stand  very 
near  it,  and  to  approach  nearer  each  other,  the  further  the  hne 
passes  from  the  blazed  trees.  Due  care  must  ever  be  taken  to 
have  the  lines  so  well  marked  as  to  be  readily  followed." 

/ 

(5G6)  illarkiug  Corners.  "  xVfter  a  true  coursing,  and  most 
exact  measurements,  the  corner  boundary  is  the  consummation  of 
the  work,  for  which  all  the  previous  pains  and  expenditure  have 
been  incurred.  A  boundary  corner,  in  a  timbered  country,  is  to 
be  a  tree,  if  one  be  found  at  the  precise  spot ;  and  if  not,  a  post  is 
to  be  planted  thereat :  and  the  position  of  the  corner  post  is  to  be 


PART  XII. ]  .Ifarking  Corners.  373 

indicated  by  trees  adjacent,  (called  Beavin^  trees)  the  angular 
bearings  and  distances  of  Avhieh  from  the  corner  are  facts  to  be 
ascertained  and  registered  in  your  field  book. 

In  a  region  where  stone  abounds,  the  corner  boundary  will  be  a 
email  monument  of  stones  along  side  of  a  single  marked  stone,  for 
a  township  corner  —  and  a  single  stone  for  all  other  corners. 

In  a  region  where  timber  is  not  near,  nor  stone,  the  corner  will 
be  a  mound  cf  earth,  of  prescribed  size,  varying  to  suit  the  ca.se. 

Corners  are  to  be  fixed,  for  township  boundaries  at  intervals  of 
every  six  miles ;  for  section  boundaries  at  intervals  of  every  mile, 
or  80  chains ;  and,  for  quarter  section  boundaries  at  intervals  of 
every  half  mile,  or  40  chains. 

Meander  Corxer  Posts  are  to  be  planted  at  all  those  points 
where  the  township  or  section  lines  intersect  the  banks  of  such 
rivers,  lakes,  or  islands,  as  are  by  law  directed  to  be  meandered," 
as  explained  in  Art.  (564). 

When  posts  are  used,  their  lengtn  and  size  must  be  propor- 
tioned to  the  importance  of  the  corner,  whether  township,  section, 
or  quarter-section,  the  first  being  at  least  24  inches  above  ground, 
and  3  inches  square. 

^Yhere  a  township  post  is  a  corner  common  to  ybi<r  townships, 

it  is  to  be  set  in  the  earth  diagonally/,  thus :  w  ^  e,  and  the  cardi- 

s 
nal  points  of  the  compass  are  to  be  indicated  thereon  by  a  cross 
line,  or  wedge,  (one-eighth  of  an  inch  deep  at  least),  cut  or  sawed 
out  of  its  top,  as  in  the  figure.  On  each  surface  of  the  post  is  to 
be  marked  the  number  of  the  particular  township,  and  its  range, 
which  it  faces.  Thus,  if  the  post  be  a  common  boundary  to  four 
townships,  say  one  and  tivo,  south  of  the  base  line,  of  range  one, 
west  of  the  meridian  ;  also  to  townships  one  and  two,  south  of  the 
base  line,  of  range  two,  west  of  the  meridian,  it  is  to  be  marked  thus : 

The  position  of  the  post  which 
From  N.  to  E.  ^  T.     1  S.    ^  is  here  taken  as  an  example,  is 

shewn  in  the  following  diagram. 
(  2  W.  ) 

fion:  N.  to  W. 

from  E.  to  S. 

frOTI  W.  to  S. 


R.  2  W. 

T.  1  S. 

R.  1  W. 

T.  1  S. 

36 

31 1 

1         6 

R.  2  W 

T.  2  S. 

R.  1  W. 

T.  2  S. 

374  r.  S.  PIBLIC  LANDS.  [vaw.  kh 

These  marks  are  to.be  distinctly  and  neatly  cliiselled  into  the 
vvood,  at  least  the  eighth  of  an  inch  deep  ;  and  to  be  also  marked 
with  red  chalk.  The  mimher  of  the  sections  which  they  respec- 
tively/ace,  will  also  be  marked  on  the  toAvnship  post. 

Section  or  mile  posts,  being  corners  of  sections,  when  they  are 
common  to  four  sections,  are  to  be  set  diagonally  in  the  earth, 
(in  the  manner  provided  for  township  corner  posts),  and  with  a 
Bimilar  cross  cut  in  the  top,  to  indicate  the  cardinal  points  of  the 
compass ;  and  on  oach  side  of  the  squared  surfaces  is  to  be  marked 
the  appropriate  number  of  the  particular  one  of  i\\efour  sections, 
respectively,  which  such  side/«ces;  also  on  one  side  thereof  are  to 
be  marked  the  numbers  of  its  toivnsMp  and  range;  and  to  make 
such  marks  yet  more  conspicuous,  (in  manner  aforesaid),  a  streak 
of  red  chalk  is  to  be  applied. 

In  the  case  of  an  isolated  township,  subdivided  into  thirty-six 
sections,  there  are  twenty-five  interior  sections,  the  south-west  cor- 
ner boundary  of  each  of  which  will  be  common  to  four  sections. 
On  all  the  extreme  sides  of  an  isolated  township,  the  outer  tiers  of 
sections  have  corners  common  6nly  to  two  sections  then  surveyed. 
The  posts,  however,  must  be  planted  precisely  like  the  former,  but 
presenting  two  vacant  surfaces  to  receive  the  appropriate  marks 
when  the  adjacent  survey  may  be  made. 

A  quarter-section  or  half-mile  post  is  to  have  no  other  mark  on 
it  than  \  S.,  to  indicate  what  it  stands  for. 

Township  corner  posts  are  to  be  notched  with  six  notches  on 
each  of  the  four  angles  of  the  squared  part  set  to  the  cardinal 
points. 

All  mile  posts  on  to-wnship  lines  must  have  as  many  notches  on 
them,  on  two  opposite  angles  thereof,  as  they  are  miles  distant 
from  the  township  corners,  respectively.  Each  of  the  posts  at  the 
corners  of  sections  in  the  interior  of  a  township  must  indicate,  by 
a  number  of  notches  on  each  of  its  four  corners  directed  to  the 
cardinal  points,  the  corresponding  number  of  miles  that  it  stands 
from  the  outlines  of  the  township.  The  four  sides  of  the  post  will 
indicate  the  number  of  the  section  they  respectively  face.  Should 
a  tree  be  found  at  the  place  of  any  corner,  it  will  be  marked  and 
Qotched,  as  aforesaid,  and  answer  for  the  corner  in  lieu  of  a  post ; 
the  kind  of  tree  and  its  diameter  being  given  in  the  field-notes. 

The  position  of  all  corner  posts,  or  corner  trees  of  Avhatever 
description,  which  may  be  established,  is  to  be  perpetuated  in  the 
following  manner,  viz :  From  such  post  or  tree  the  courses  shall  be 
taken,  and  the  distances  measured,  to  two  or  more  adjacent  trees, 
m  opposite  directions,  as  nearly  as  may  be,  which  are  called 
^Bearing  trees,'  and  are  to  be  blazed  near  "the  ground,  with  a  large 
blaze  facing  the  post,  and  having  one  notch  in  it,  neatly  and  plainly 


PART  XII. ]  Marking  Coruers.  375 

made  -uith  an  axe,  square  across,  and  a  little  below  the  middle  ol 
the  blaze.  The  kind  of  tree  and  the  diameter  of  each  are  facts  to 
be  distinctly  set  forth  in  the  field-book. 

On  each  bearing  tree  the  letters  B.  T.,  must  be  distinctly  cut 
into  the  wood,  in  the  blaze,  a  little  above  the  notch,  or  on  the  bark, 
with  the  number  of  the  range,  township,  and  section. 

At  all  township  corners,  and  at  all  section  corners,  on  range  or 
township  lines,  four  bearing  trees  are  to  be  marked  in  this  manner, 
one  in  each  of  the  adjoining  sections. 

At  interior  section  corners  four  trees,  one  to  stand  within  each 
of  the  four  sections  to  which  such  corner  is  common,  are  to  be 
marked  in  manner  aforesaid,  if  such  be  found. 

From  quarter  section  and  meander  corners  two  bearing  trees 
are  to  be  marked,  one  within  each  of  the  adjoining  sections. 

Stones  at  township  corners  (a  small  monument  of  stones  being 
alongside  thereof)  must  have  six  notches  cut  with  a  pick  or  chisel 
on  each  edge  or  side  towards  the  cardinal  points  ;  and  where  used 
as  section  corners  on  the  range  and  township  Unes,  or  as  section 
corners  in  the  interior  of  a  township,  they  will  also  be  notched  by 
a  pick  or  chisel,  to  correspond  with  the  directions  given  for  notch- 
ing posts  similarly  situated. 

Stones,  when  used  as  quarter-section  comers,  wUl  have  ^  cut 
on  them ;  on  the  west  side  on  north  and  south  lines,  and  on  the 
north  side  on  east  and  west  Unes. 

Whenever  bearing  trees  are  not  found,  mounds  of  earth,  or 
stone,  are  to  be  raised  around  jyosts  on  which  the  corners  are  to 
be  marked  in  the  manner  aforesaid.  Wherever  a  mound  of  earth 
is  adopted,  the  same  will  present  a  conical  shape  ;  but  at  its  base, 
on  the  earth's  surface,  a  quadrangular  trench  will  be  dug;  a  spade 
deep  of  earth  being  thrown  up  from  the  four  sides  of  the  line,  out- 
side the  trench,  so  as  to  form  a  continuous  elevation  along  its  outer 
edge.  In  mounds  of  earth,  common  to  four  townships  or  to  four 
sections,  they  AviU  present  the  angles  of  the  quadrangular  trench 
(diagonally^  towards  the  cardinal  points.  In  mounds  common 
only  to  two  townships  or  two  sections,  the  sides  of  the  quadrangular 
trench  will /ace  the  cardinal  points. 

Prior  to  piling  up  the  earth  to  construct  a  mound,  in  a  cavity 
formed  at  the  corner  boundary  point  is  to  be  deposited  a  stone,  or 
a  portion  of  charcoal,  or  a  charred  stake  is  to  be  driven  twelve 
inches  down  into  such  centre  point,  to  be  a  tvitness  for  the  future. 
The  surveyor  is  farther  specially  enjoined  to  plant  midway 
between  each  pit  and  the  trench,  seeds  of  some  tree,  those  of  fruit 
trees  adapted  to  the  chmate  being  always  to  be  preferred. 

Double  corners  are  to  be  found  nowhere  except  on  the  Standard 
Parallels  or  Correction  lines,  whereon  are  to  appear  both  the  cor 


376  r.  S.  PUBLIC  LANDS.  [pakt  xri 

ners  whlcli  mark  the  intersections  of  the  lines  which  close  theieon. 
and  those  from  which  the  surveys  start  in  the  opposite  direction. 

The  corners  which  are  established  on  the  standard  parallel,  at 
the  time  of  running  it,  are  to  be  known  as  '  Standard  Corner%^ 
and,  in  addition  to  all  the  ordinary  marks,  (as  herein  prescribed), 
they  will  be  marked  with  the  letters  S.  C.  The  '  closing  eorneri' 
will  be  marked  C.  C." 

(567)  Field  Books.  There  should  be  several  distinct  and  sepa- 
rate field-books ;  viz. : 

"  1.  Field-notes  of  the  meridian  and  base  lines,  showing  the 
establishment  of  the  township,  section  or  mile,  and  quarter-section 
or  half-mile,  boundary  corners  thereon ;  with  the  crossings  of 
streams,  ravines,  hills,  and  mountains ;  character  of  soil,  timber, 
minerals,  &c.  These  notes  will  be  arranged,  in  series,  by  mile 
stations,  from  number  07ie  to  number . 

2.  Field-notes  of  the  '  standard  parallels,  or  correction 
lines,'  showing  the  establishment  of  the  township,  section,  and 
quarter-section  corners,  besides  exhibiting  the  topography  of  the 
country  on  line,  as  required  on  the  base  and  meridian  lines. 

3.  Field-notes  of  the  exterior  lines  of  townships,  showing  the 
establishment  of  the  corners  on  line,  and  the  topography,  as  aforesaid. 

4.  Field  notes  of  the  subdivisions  of  townships  into  sections 
and  quarter-sections ;  at  the  close  whereof  will  follow  the  notes  of 
the  MEANDERS  of  navigable  streams.  These  notes  will  also  show, 
by  ocular  observation,  the  estimated  rise  and  fall  of  the  land  on 
the  line.  A  description  of  the  timber,  undergrowth,  surface,  soil, 
and  minerals,  upon  each  section  line,  is  to  follow  the  notes  thereof, 
and  not  to  be  mixed  up  with  them.^' 

5.  The  "  Geodetic  Field-book,"  comprising  all  triangulations, 
angles  of  elevation  and  depression,  levelling,  &c. 

The  examples  on  the  next  two  pages,  taken  from  the  "  Insti-up- 
tions "  which  we  have  followed  throughout,  will  shew  what  is 
required. 

The  ascents  and  descents  are  recorded  in  the  right-hand  columns, 


r.u;r  xn] 


Fh  l<l-i\oU'S. 


87: 


FIELD  NOTES  OF 

THE    EXTERIOR    LINES 

OF  AN  ISOLATED  TOAVNSHIP. 


Ftela  notes  of  /he  Siurrey  of  township  25  north,  of  range  2  west,  of  the  WiUameltt 
meridian,  in  the  Terriiory  o/ Oregon,  by  Robert  Acres,  deputy  surveyor,  under  his 
contniet  No.  1,  bearing  date  the  2d  day  of  January,  IS.jI. 


t'Chs.    Iks, 


E;ist. 


West 

40.00 


G2.50 
80.00 


West. 
40.00 


65.00 
80  00 


West 
40. OC 


80.00 


Township  lines  commenced  January  20,  1851. 

Soiuhern  bouiiclary  variation  18°  41'  E. 

On  a  random  line  on  the  sonlh  honiidaries  of  sections  il,  32, 
33,  34,  35,  and  36.  Set  temporary  mile  and  half'-inile  po.sl.v 
and  intersected  the  eastern  boundary  2  chains  20  links  noril 
of  the  true  corner  5  miles  74  chains  53  links. 

Therefore  the  correction  will  be  5  chains  47  links  W.  37.1 
links  S.  per  mile.  


Truk  southern  boundary  variation   18°  41'  E. 
On  the  southern  boundary  of  sec.  36,  Jan.  24,  1851. 
Set  qr.  sec.  post  from  which 

a  beech  24  iu.  dia.  bears  N.  11  E.  38  Iks.  dist. 

a    do        9      do  do     S.    9  E.  17       do 

a  brook  8  1.  wide,  course  NW 

Set  post  cor.  of  sees.  35  &  36,  1  &  2,  from  which... 

a  beech      9  in.  dia.  bears  S.  46  E.      8  1.  dist. 

a      do         8     do  do     S.  62  W.     7     do 

a  W.  oak  10     do  do    N.  19  W 

a  B.  oak  14     do  do    N.  29  E. 

Land  level,  part  wet  and  swampy  ; 

hickory,  &.c. 


14  do 
16  do 
timber  beech,  oak,  ash 


On  the  S.  boundary  of  sec.  35 — 

Set  qr.  sec.  post,  with  trench,  from  which 

a  beech  6  in.  dia.  bears  N.  80  E.  8  1.  dist. 

planted  SW.  a  yellow  locust  seed. 

To  beginning  of  hill 

Set  post,  with  trench,  cor.  of  sees.  34  &  35,  2  &  3,  from  vvhicl 

a  beech  10  in.  dia.  bears  S.  51  E.  13  1.  dist. 
do       10     do  do    N.  56  W.  9     do 

planted  SW.  a  white  oak  acorn, 
NE.  a  beech  nut. 
Land  level,  rich,  and  good   for  farming;   timber  same. 


On  the  S.  boundary  of  sec.  34 — 

Set  qr.  sec.  post,  with  trench,  from  which 

a  B.  oak  10  in.  dia.  bears  N.  2  E.  635  1.  dist. 

Planted  SW.  a  beech  nut. 
To  corner  of  sections  33,  34,  3  and  4,  drove  charred  stake; 

raised  mound  with  trench  as  per  instructions,  and 

Planted  NB.  a  W.  oak  ac'n  ;  NW.  a  yel.  locust  seed. 
SE.  a  butternut ;  SW^  a  beech  nut 
Land   level,  rich  and  good  for  farming,  some  scattering  oal^ 

and  walnut. 


Fset. 


a  10 


d  10 

a    .J 


a  II) 


a    5 
s20. 


a    5 


10 


&c.,         &c.. 


&c 


S78 


U.  S.  PUBLIC  LVXDS. 


PART  xri. 


FIELD  NOTES  OF  THE 

SUBDIVISIONAL  OR  SECTIONAL  LINES, 

AND    ME.INDERS. 


Township  25  iV.,  Range  2   \V.,   WUlamette  Mer. 


Cbs.  lk= 

Nortli. 

9.19 

29.97 

40.00 


51.90 
76.73 
80.00 


En  St. 
9.00 
15.00 
40.00 
55.00 
72.00 
80.00 


.Subdivisions.     Commenced  Ffbnmiy  1,  1851. 
Between  sees.  35  and  36 — 

A  beech  30  in.  dia 

A  beech  30  in.  dia 

Set  qr.  sec.  post,  from  which 

a  beech  15  in.  dia.  bears  S.  48  E.   12  1.  dist. 

a    do       8      do         do    N.  23  VV.  45    do 

A  beech  18  in.  dia 

A  sugar  30  in.  dia , 

Set  a  post  cor.  of  sees.  25,  26,  35,  36,  from  which 

a  beech    24  in.  dia.  bears  N.  62  W.  17  1.  dist. 


a  poplar  36     do 
a      do      20     do 
a  beech    28     do 
Land   level,  second 
iind'sr.  spice,  &c. 


do      S.  66  E.    34     do. 
do      S.  70  W.  50     do. 

do  N.  60  E.    45     do. 

rate  ;  limber  beech,  poplar,  sugar,  and 


On  random  line  between  sees.  25  aad  itO — 

A  brook  30  1.  wide,  course  N , 

To  foot  of  hill , 

Set  tempoi'ary  qr.  sec.  post 

To  opposite  foot  of  hill _ 

A  hriiok  15  1.  wide,  course  N 

Inter.sect  E.  boundary  at  post 

Laiui  level,  second  rate;   timber,  beech,  oak,  ash,  &c 


&c.,  &c., 


ike. 


FeeU 


d  10 
d  5 
d    5 


d  5 
d  8 
d    2 


d  10 
d  10 
a  60 
d  40 
d20 
a  10 


Meanders  of  Chickeeles  River. 

Beginning  at  a  meander  post  in  the  northern  township  boundary,  and  thonce  on 
the  left  bank  dowa  stream.      Commenced  February  11,  1851. 


Courses. 

Dist. 
Ch.'=.   Iks. 

REMARKS. 

S.  76  W. 
S.  61   W. 
S.  61  VV. 

18.46 

10.00 

8.18 

10.69 
5..59 
8.46 
16.50 
21.96 
27  53 

Ill  Section  4  bearing  to  corner  sec.  4  on  right  bank  N.  70° 
Bearing  lo  cor.  see.  4  and  5,  right  bank  N.  52°  W. 
To  post  in  line  between  sections  4  and  5,  breadth  of  river 
triansnlation  9  chains  51  links. 

VV 

by 

S.  54  W. 
S.  40  VV. 
S.  50  VV. 
S.  37  VV. 
S.  44  VV. 
S.  36  W. 

In  Section  5. 

To  upper  corner  of  John  Sm  th's  (;laim,  course  E. 

To  post  in  line  between  sections  5  and  8,  breadth  of  rivei 
trian^ulation  8  chains  78  links. 

:>^ 

&c.,         &c            &c. 

APPENDIX. 


APPENDIX    A. 


SYNOPSIS  OF  PLANE  TRIGONOMETRY. 


(1)  DefillitJOll.  Plaae  Trigonometry  is  that  branch  of  Mathematical 
Science  which  treats  of  the  relations  between  the  sides  and  angles  of  plane  trian- 
gles. It  teaches  how  to  find  any  three  of  these  six  parts,  when  the  other  three 
are  given  and  one  of  them,  at  least,  is  a  side. 

(2)  Angles  a,nd  Arcs.  The  angles  of  a  triangle  are  measured  by  the 
arcs  described,  with  any  radius,  from  the  angular  points  as  centres,  and  intercepted 
between  the  legs  of  the  angles.  These  arcs  are  measured  by  comparing  them  with 
an  entile  circumference,  described  with  the  same  radius.  Every  circumference  is 
regarded  as  being  divided  into  360  equal  parts,  called  degrees.  Each  degree  is  di- 
vided into  60  equal  parts,  called  minutes,  and  each  minute  into  60  seconds.  These 
divisions  are  indicated  by  the  marks  °  '  ".  Thus  28  degi-ees,  17  minutes,  and  49 
seconds,  are  written  28°  17'  49".  Fractions  of  a  second  are  best  expressed  deci- 
mally. An  arc,  including  a  quarter  of  a  circumference  and  measuring  a  right 
angle,  is  therefore  90°.  A  semieircumference  comprises  180°.  It  is  often  repre- 
sented by  jr,  which  equals  3.14159,  <fec.,  or  3^  approximately,  the  radius  being  unity. 

The  length  of  1°  in  ppits  of  radius  =  0.01745329  ;  that  of  1 '=  0.00029089  ;  and 
that  of  1"=0.000004&6. 

The  length  of  the  radius  of  a  circle  in  degrees,  or  360ths  of  the  circumference 
=  57°.29578  =  57°  17'  24".8  =  3437'.747  =  206264' '.S.-l- 

An  arc  may  be  regarded  as  generated  by  a  point,  M, 
moving  from  an  origin.  A,  around  a  circle,  in  the  direction 
of  the  arrow.  The  point  may  thu?  describe  arcs  of  any 
lensrths,  such  as  AJVI ;  AB  =  90°  =  i  tt  ;  ABC  =  180°  =  ^r ; 
ABCD  =  270°  =  f  TT ;  ABCDA  =  360°  =  2  «-. 

The  point  may  still  continue  its  motion,  and  generate 
arcs  greater  than  a  circumference,  or  than  two  circum- 
ferences, or  than  three ;  or  even  infinite  in  length. 

While  the  point,  M,  describes  these  arcs,  the  radius, 
OM,  indefinitely  produced,  generates  corresponding  angles. 


♦  For  merely  solving  triangles,  only  Articles  (1),  (2),  (3),  (5),  (6\  (10),  (11),  and  (12),  are  needwl 
r  Tbe  number  of  seconds  in  any  arc  which  is  given  in  parts  of  radius,  radius  being  unity,  equals 

the  length  of  the  arc  so  given  divided  by  the  length  of  the  arc  of  one  second;  or  multiplied  by  th« 

mmber  of  seconds  in  radios. 


380 


TRIGOXOIIETRF. 


[app. 


Fig.  893. 


If  the  point,  M,  shculJ  move  from  the  origin,  A,  in  the  contrary  directiou  to  its 
former  movement,  the  arcs  generated  by  it  are  regarded  as  negative,  or  minux , 
and  so  too,  of  necessity,  the  angles  measured  by  the  arcs. 

Arcs  and  angles  may  therefore  vary  in  length  from  0  to  +  qo  in  one  direction, 
and  from  0  to  —  co  in  the  contrary  direction. 

The  Complement  of  an  arc  is  the  arc  which  would  remaiu  after  subtracting  the 
arc  from  a  quarter  of  the  circumference,  or  from  90°.  If  the  arc  be  more  than  9U°, 
its  complement  is  necessarily  negative. 

The  Supplement  of  an  arc  is  what  would  remain  after  subtracting  it  from  hall 
fhe  circumference,  or  from  180°.  If  the  arc  be  more  than  180°,  its  supplement 
is  necessarily  negative. 

(3)  Trigonometrical  Lines.  The  relations  of  the  sides  of  a  triangie 
to  its  angles  are  what  is  required ;  but  it  is  more  convenient  to  replace  the  angles 
by  arcs ;  and,  once  more,  to  replace  the  arcs  by  certain  straight  linos  depending 
upon  them,  and  increasing  and  decreasing  with  them,  or  conversely,  in  such  a  way 
that  the  length  of  the  lines  can  be  found  from  that  of  the  arcs,  and  vice  versd..  It 
is  with  these  lines  that  the  sides  of  a  triangle  are  compared.*  These  lines  are 
called  Trigonometrical  Lines  ;  or  Circular  Functions,  because  their  length  is  a  func- 
tion of  that  of  the  circular  arcs.  The  principal  Trigonometrical  lines  are  Sines, 
Tangents,  and  Secants.     Chords  and  versed  siues  are  also  used. 

The  SINE  of  an  arc,  AM,  is  the  perpendicular, 
MP,  let  fall,  from  one  extremity  of  the  arc,  upon 
the  diameter  which  passes  through  the  other  ex- 
tremity. 

The  TANGENT  of  an  arc,  AM,  is  the  distance, 
AT,  intercepted,  on  the  tangent  drawn  at  one 
extremity  of  the  arc,  between  that  extremity 
and  the  prolongation  of  the  radius  which  passes 
through  the  other  extremity. 

The  SECANT  of  an  arc,  AM,  is  the  part,  OT, 
of  the  prolonged  radius,  comprised  between  the 
centre  and  the  tangent. 

The  sine,  tangent,  and  secant  of  the  complement  of  ari  arc  are  called  the  Co- 
sine, Co-tangent,  and  Co-secant  of  that  arc.  Thus,  MQ  is  the  cosine  of  AM,  BS 
its  cotangent,  and  OS  its  cosecant.  The  cosine  MQ  is  equal  to  OP,  the  part  of  the 
radius  comprised  between  the  centre  and  the  foot  of  the  sine. 

The  chord  of  an  arc  is  equal  to  twice  the  sine  of  half  that  arc. 

The  versed-sine  of  an  arc,  AM,  is  the  distance,  AP,  comprised  between  the  origin 
of  the  arc  and  the  foot  of  the  sine.  It  is  consequently  equal  to  the  difference  be- 
tween the  radius  and  the  sine. 

The  Trigonometrical  lines  are  usually  written  in  an  abbreviated  form.  Calling 
the  arc  AM  =  a,  we  write, 

MP  =  sin.  a.  AT  =  tan.  a.  C^  =sec.  a. 

MQ  =  cos.  a,  BS  =»  cot.  a.  OS  =  cosec.  a. 

The  period  after  sin.,  tan.,  Ac,,  indicating  abbreviation,  is  frequently  omitted. 

The  arcs  whose  sines,  tangents,  <fce.,  are  equal  to  a  line  =a,  are  written, 
sin.       a,  or  arc  (sin.  =  a) ; 
tan.       a,  or  arc  (tan.=  a) ;  <fec. 

•  For  th«  frreat  value  of  this  '.ndirect  mode  of  comparing  the  sides  and  angles  of  triangles,  lot 
Comte'R  "  Philosophy  c  Mattematics,"  (Harpers',  1S51,)  page  225. 


A.] 


TRICO\0>IETRY. 


nsi 


(4)  The  lines  :is  ratios.    Tlie  ratios 

del  ween  the  trigonometrical  lines  and  the  radius 
are  the  same  for  the  same  angles,  or  number  of 
degrees  in  an  arc,  whatever  the  length  of  *.he  ra- 
dius or  arc.  Consequently,  radius  be.ng  unity, 
these  lines  may  be  expressed  as  simple  ratios. 
Thus,  in  the  right-angled  triangle  ABC,  we 
would  have 


Fig.  899. 


sin.  A  = 


BC opposite  side 

AB        hypothenuse ' 

BC opposite  side 

AC        adjacent  side' 


AB hypothenuse 

AC        adjacent  side 


cos.  A  =  - 


' adjacent  side 


tan.  A: 


AB 

cot.  A  =  —  ■ 
BC 

.       AB 

30SCC.    A  =  _: 


hypothenuse  ' 

adjacent  side 
opposite  side ' 

hypothenuse 
opposite  side' 


When  the  radius  of  the  arcs  which  measure  the  angles  is  unity,  these  ratios  may 
be  used  for  the  lines.  If  the  radius  be  any  other  length,  the  results  which  have 
been  obtained  by  the  above  suppo.sition,  must  be  modified  by  dividing  each  of  the 
trigonometrical  lines  in  the  result  by  radius,  and  thus  rendering  the  equations  of 
the  results  "  homogeneous."  The  same  effect  would  be  produced  by  multiplying 
each  term  in  the  expression  by  such  a  power  of  radius  as  would  make  it  contain  a 
number  of  linear  factors  equal  to  the  greatest  number  in  any  term.  The  radius 
is  usually  represented  by  r,  or  R. 

(5)  Their  variations  in  length.  As  the  point  M  moves  around 
the  circle,  and  the  arc  thus  increases,  the  sines,  tangents,  and  secants,  starting 
from  zero,  also  increase ;  till,  when  the 
point  M  has  arrived  at  B,  and  the  arc  has 
become  90°,  the  sine  has  become  equal  to 
radius,  or  unity,  and  the  tangent  and  se- 
cant have  become  infinite.  The  comple- 
mentary lines  have  decreased ;  the  co- 
sine being  equal  to  radius  or  unity  at 
starting  and  becoming  zero,  and  the  co- 
tangent and  cosecant  pa.«sing  from  infin- 
ity to  zero.  When  the  point  M  has 
passed  the  first  quadrant  at  B  and  is 
proceeding  towards  C,  the  sines,  tan- 
gents, and  secants  begin  to  decrease,  till, 
when  the  point  has  reached  C,  they  have 

the  same  values  as  at  A.     They  then  begin  to  increase  again,  and  so  on. 
Table  on  page  382  indicates  these  variations. 

The  sines  and  tangents  of  very  small  arcs  may  be  regarded  as  sensibly  propor- 
tional to  the  ares  themselves  ;  so  that  for  sin.  a",  we  may  write  a.  sin.  1"  ;  and 
similarly,  though  less  accurately,  for  sin.  a,  we  may  write  a .  sin.  1'. 

The  sines  and  tangents  of  very  small  arcs  may  similarly  be  regarded  as  sensibly 
of  the  same  length  as  the  arcs  themselves.* 


The 


•  Consequently,  the  note  on  page  379  may  read  thus :  The  number  of  seconds  in  any  very  small 
•re  given  in  parts  of  radius,  radius  be'-ig  unity,  is  equal  to  the  length  of  the  arc  so  given  divided 
by  sin.  1 


382 


TRIG0\0.1IETRY. 


lAPP.    A, 


a  being  the  length  of  any  arc  expressed  in  parts  of  radius,  the  lengths  of  its  bIm 
and  cosine  may  be  obtained  by  the  following  series : 


sm.  a=a  — 


2.3 


+  ? 


2.3.4.5        2.3. 


—  +,  etc. 


cos.  a=l ^4" 


+,  etc 


2    3.4       2 6 

Let  it  be  required  to  find  cos.  30°,  by  the  above  series. 

30 
30°  =  —     »r  =  |  X  3.1416  =.5236. 
180 

Substituting  this  number  for  a,  the  series  becomes,  taking  only  three  terms  of  it, 

(.5236)'^  ,    (.5236)' 
1  —  ^  '  +  ^  '  -,  etc.  =  1  -  0.137078  +  0.003130  —  .866052  ; 

•which  is  the  correct  value  of  cos.  30°  for  the  first  four  places  of  decimals. 

The  lengths  of  the  other  lines  can  be  obtained  from  the  mutual  relations  given 

in  Art.  (7.)     Some  particular  values  are  given  below. 

sin.  30°  =  i.  sin.  45°  =  i^/2.  .sin.  60°  =  ^^3 

tan.  30°  =  i^/.3.  tan.  45°  =  1.  tan.  60°  =  ^3. 

sea  30°  =  §v/.3.  sec.  45°  =  v^2.  sec.  60°  =  2. 

(6)  Tlieir  chang'CS  of  sis'!!.  Lines  measured  in  contrary  direeticns 
from  a  common  origin,  usually  receive  contrary  algebraic  signs.  If  then  all  the 
lines  in  the  first  quadrant  are  called  positive,  their  signs  will  change  in  some  of 
the  other  quadrants.  Thus  the  sines  in  the  first  quadrant  being  all  measured  up- 
ward, when  they  are  measured  downward,  as  they  are  in  the  third  and  fourth 
quadrants,  they  will  be  negative.  The  cosines  in  the  first  quadrant  are  meas- 
ured from  left  to  right,  and  when  they  are  measured  from  right  to  left,  as  in  the 
second  and  third  quadrants,  they  will  be  negative.  The  tangents  and  secants  fol- 
low similar  rules. 

The  variations  in  length  and  the  changes  of  sign  are  all  indicated  in  the  follow- 
ing table,  radius  being  unity.  The  terms  "  increasing"  and  "  decreasing"  apply  to 
the  lengths  of  the  hues  without  any  reference  to  their  signs. 

Lengths  and  Signs  of  the  Trigonometrical  Lines  for  Arcs  from  0^  to  360° 


Arcs. 

0° 

Between  0°  and  90°. 

90O 

Between  90°  and  ISQC. 

130O 

Sine     .     .     . 
Tangent   .     . 
Secant      .    . 
Cosme      .     . 
Cotangent     . 
Cosecant  .     . 

0 
0 

+1 
+1 
±^ 

d=oo 

-f-,  and  increasing, 
-(-,  and  increasing, 
+,  and  increasing, 
+,  and  decreasing, 
-f,  and  decreasing, 
-(-,  and  decreasing. 

+  1 

±C0 

±co 
0 
0 

+  1 

-f-,  and  decreasing, 
— ,  and  decreasing, 
— ,  and  decreasing, 
— ,  and  increasing, 
— ,  and  increasing, 
-f-,  and  increasing. 

0 
0 

-1 

-1 

=Foo 

±00 

Arcs. 

ISOO 

Between  130°  and  270°.      270° 

Between  270°  and  360^. 

360° 

Sine     .     .     . 
Tangent 
Secant 
Cosine .     .     . 
Cotangent     . 
Cosecant  .     . 

0 
0 

—1 
-1 

±» 

— ,  and  increasing, 
+,  and  increasing, 
— ,  and  increasing, 
— ,  and  decreasing, 
-f-,  and  decreasing, 
— ,  and  decreasing. 

-1 

±00 

T^ 

0 
0 

-1 

— ,  and  decreasing, 
— ,  raid  decreasing, 
+,  and  decreasing, 
4",  and  increasing, 
— ,  and  increasing, 
— ,  and  increasing. 

0 
0 

+  1 

+  1 

q^oo 

Ifoo 

i^pp.  a]  TRIGOXOIETRY.  3pj 

From  this  table,  and  Fig.  400,  we  see  that  an  arc  and  its  ^upplerient  have  the 
same  sine ;  and  that  their  tangents,  secants,  cosines,  and  cotangents  are  of  equal 
length  but  of  contrary  signs ;  while  the  cosecants  are  the  same  in  both  lenijth 
»nd  sign. 

We  also  deduce  from  the  figure  the  following  consequences : 

sia  (a°+  180°)  =-sin.  a°.  cos.  (a°4-  180°)  =-cos.  a°. 

tan.  (a°+  180°)  =  tan.  a°.  cot.  la°-\-  180°)  =  coL  a', 

sec.  (a°-f  180°)  =— sec.  a°.         cosec.  {a°-\- 180°)  =— cosec.  a°. 

sin.  (— a°)=— sin.  a°.  cos.  (  — a°)  =  cos.  o°. 

tan.  (— a°)  *=— tan.  a°.  cot.  (— a°)  =— cot.  a°. 

sec  (— a°)  =  sec.  a°.  cosec.  (— a°)  =— cosec.  a°. 

An  infinite  number  of  arcs  have  the  same  trigonometrical  lines ;  for,  an  are  a, 
the  same  arc  plus  a  circumference,  the  same  arc  plus  two  circumferences,  and  so 
on,  would  have  the  same  sine,  <fec. 

"  To  bring  back  to  the  first  quadrant"  the  trigonometrical  lines  of  any  large  arc, 
proceed  thus:  Let  1029°  be  an  arc  the  sine  of  which  is  desired.  Take  from  it  as 
many  times  360°  as  possible.     The  remainder  will  be  309°.     Then  we  shall  have 

8in.309°=sin.(180°— 309°)=sia-129°=-sin.l29°=-sin.(180°  — ^''9°)=— sin.51' 

(7)  Their  mutual  relations.     Radius  being  unity, 

tan.  a° 

COS.  a'  sm.  a" 

1  o  1 

cosec.  a   =■ 


COS.  a  sin.  a 

tan.  a° Xcot.  a°  =  1.  (sia  a°)'  +  (cos.  a°)»  =  1  * 

1  +  (tan.  ay  =  (sea  a°)'.  1  +  (cot.  a°y  =  (cosec.  a°)«. 

Hence,  any  one  of  the  trigonometrical  lines  being  given,  the  rest  can  be  found 
from  some  of  these  equations. 

(§)  Two  arcs.     Let  a  and  b  represent  any  two  arcs,  a  being  ilie  greater 
Then  the  following  formulas  apply : 

sin.  {a-\-b)  =  sin.  a .  cos.  b  -(-  cos.  a .  sin.  b. 
sia  (a  —  ^)  ^=  sin.  « .  cos.  b  —  cos.  a .  sia  b. 
cos.  (a  -j-  6)  =  COS.  a .  cos.  b  —  sin.  a .  sia  b. 
cos.  (a  —  6)  =  COS.  a .  cos.  b  +  sin.  a .  sin.  6. 

.        /      I    ,  X         tan.  a  +  tan.  b 

tan.  (a  +  6)  =  :; 

1  —  tan.  a .  tan.  6 

tan.  a—  tan.  b 

tan.  (a  —  6)  =  — -. 

1  -\-  tan.  a .  tan.  o 

,    ,     ,    ,  ^       cot.  a .  cot.  6  —  1 

cot.  (a  -f  6)  =: J—. . 

cot.  6  -|-  cot  a 

^   ,        ,  ^       cot.  a .  cot.  6  +  1 

cot.  (a  —  b)  = ; ' — . 

cot.  6  —  cot.  a 

•  The  square,  &c.,  of  the  sine,  &c,  of  an  arc,  is  often  expressed  by  placing  the  exponent  between 
the  abbreviation  of  the  name  of  the  trigonometrical  line  and  the  number  of  the  degrees  in  the  arc 
<hu3,  sin.'  a°,  tan.^  a°,  &c.    But  the  notation  given  above,  places  the  index  as  used  by  6ans^ 
telambre,  Arbogast,  &c.,  though  the  first  two  omit  the  parentheses. 


334  TRIGONOMETRY.  [app.  a. 

Bin.  a  .  sin.  b=^i.  cos.  (a  —  b)  —  i  cos.  (a  +  b). 

COS.  a .  COS.  6  =:  i .  COS.  (a  +  6)  +  i  cos.  (a  —  i). 

sin.  o .  COS.  6  =  ^ .  sin.  {a-\-  b)-\-i  sin.  (a  —  6). 

cos.  a .  sin.  6  =  i .  sin.  (a  +  &)  —  i  sin.  (a  —  b). 
sin.  a  4- sin.  6  :=  2  sin.  -J-  (a  +  6)  cos.  i  (a  —  b). 
cos.  a  +  COS.  6  =  2  cos.  -J-  {a-{-  b)  cos.  1  (a  —  6). 
sin.  a  —  sin.  6  =  2  sin.  ^  (a  —  6)  cos.  ^  (a  +  6). 
COS.  6 .—  cos.  a  =  2  sin.  -J  (a  —  6)  sin.  i  (a  +  6). 


,  u.  -|-   ia,u.  1/ 

cos. 

a. 

COS.  6' 

tan. 

a  —  tan.  6 

sin. 
cos. 

{a 
a . 

-6) 

COS.  6' 

cot. 

6  +  cot.  a 

sin. 
sin. 

{a 

a . 

sin.  6' 

cot. 

b  —  cot.  a 

sin. 

_(a 

-6) 

sin.  a .  sin,  6 

(9)  Double  and  Iia.lf  arcs.     Letting  a  represent  any  arc,  as  befoi« 
we  have  the  following  formulas  : 
sin.  2  a  =:  2  sin.  a .  cos.  a. 

COS.  2  a  =  (cos.  a)*  —  (sin.  a)'  =  2  (cos.  a)"  —  1  =  1  —  2  (sin.  a)*, 
2  tan.  a  2  cot,  a  2 


tan.  2  a  = 


1  —  (tan.  af      (cot.  af  —  1       cot.  a  —  tan.  a 


^   „  (cot.  a)"  -  1        ,  ,    ^ 

cot.  2  a  ^ =  A  (cot.  a  —  tan.  a). 

2  cot.  a  ^  ^  '^ 

sin.  i  a  =  ^  [  ^  (1  —  cos  a)  ]. 

cos.  •|a  =  ^  [i(l  +  COS.  a)  ]. 

sin.  a          1  —  COS.  a         ,  /I  —  cos.  a\ 
tan.  i  a  =  — ; ^ : =  »^  I  — I. 

1  4"  COS.  a  sm.  a  \1  +  cos.  a/ 

1  +  COS.  a  sin.  a  .  /I  +  cos.  a\ 

cot.  -J  a  =  — ^ = =  v/  I  — ■ I- 

sin.  a  1  —  COS.  a  \1  —  cos.  a/ 

(10)  Trig^onometrical  Tables.  In  the  usual  tables  of  the  natural 
Trigonometrical  lines,  the  degrees  from  0°  to  45°  are  found  at  the  top  of  the  table, 
and  those  from  45°  to  90°  at  the  bottom ;  the  latter  being  complements  of  the 
former.  Consequently,  the  columns  which  have  Sine  and  Tangent  at  top  have 
Cosine  and  Cotangent  at  bottom,  since  the  cosine  or  cotangent  of  any  arc  is  the 
same  thing  as  the  sine  or  tangent  of  its  complement.  The  minutes  to  be  added 
to  the  degrees  are  found  in  the  left-hand  column,  when  the  number  of  degrees  at 
the  top  of  the  page  are  used,  and  in  the  right-hand  column  for  the  degrees  when 
at  the  bottom  of  the  page.  The  lines  for  arcs  intermediate  between  those  in  the 
tables  are  found  by  proportion.  The  lines  are  calculated  for  a  radius  equal  unity. 
Hence,  the  values  of  the  sines  and  cosines  ai'e  decimal  fractions,  though  the  point 
is  usually  omitted.  So  too  are  the  tangents  from  0°  to  45°,  and  the  cotangent* 
from  90°  to  45°,     Beyond  those  points  they  are  integers  and  decimals. 

The  calculations,  like  all  others  involving  large  numbers,  are  shortened  by  the 
Ofie  of  logarithms,  which  substitute  addition  and  subtraction  for  multiplication  and 
division;  but  the  young  student  should  avoid  the  frequent  error  of  regarding  logs 
f ithms  as  a  necessary  part  of  trigonometry. 


AXP.   aJ] 


TRIGONOMETRY. 


SM 


SOLUTION  OF  TRIANGLES. 

(11)    Right-ang^led    Ti*iaug:Ics.     Let 

ABC  be  any  right-angled  triangle.  Denote  the 
sides  opposite  the  angles  by  the  corresponding  small 
letters.  Then  any  one  side  and  one  acute  angle,  or 
any  two  sides  being  given,  the  other  parts  can  be  ob- 
tained by  one  of  the  following  equations : 


Flg.40L 


Given. 

Required. 

Formulas. 

a,  b 
a,  c 

a,  A 

b,  A 

c,  A 

c,  A,  B 

b,  A,  B 

b,  c,    B 

a,  c,    B 
a,  b,    B 

c  =7(a«  +  6") ;  tan.  A  =^  ;  cot.  B  =^. 
0                      6 

b  =v/(c'  —  a') ;  sin.  A  =:  -  ;  cos.  B  ^  -. 
c                     c 

b  =  a.  cot.  A ;  c  =  -Ar-  ;  B  =  90°  —  A. 
sm.  A 

n  •— h    tnn    A-   c —              •   Ti  —  Q0°...    \ 

COS.  A 
a=  c .  sin.  A ;  6  =:  c  cos.  A  ;  B  =  90°  —  A. 

(12)    Obliqiie-ang^led  Ti'iau- 

gles.  Let  ABC  be  any  oblique-angled 
triangle,  the  angles  and  sides  being  noted 
as  in  the  figure.  Then  any  three  of  its  six 
parts  being  given,  and  one  of  them  being  a 
side,  the  other  parts  can  be  obtained  by  one 
of  the  following  methods,  which  are  found- 
ed on  these  three  theorems. 

Theoeem  L — In  every  plane  triangle,  the  sines  of  the  angles  are  to  each  other  at 
the  opposite  sides. 

Theorem  II. — In  every  plane  triangle,  the  sum  of  two  sides  is  to  their  differenct 
ms  the  tangent  of  half  tlte  sum  of  the  angles  opposite  those  sides  is  to  the  tangeiit  of 
half  their  difference. 

Theorem  III. — In  every  plane  triangle,  the  cosine  of  any  angle  is  equal  to  afrao- 
tion  whose  numerator  is  the  sum  of  the  squares  of  the  sides  adjacent  to  the  angle,  mi- 
nus the  sqtiare  of  the  side  opposite  to  the  angle,  and  whose  denominator  is  twice  the 
product  of  the  sides  adjacent  to  the  angle. 

All  the  cases  for  solution  which  can  occur,  may  be  reduced  to  four. 

Case  1. — Given  a  side  and  two  angles.  The  third  angle  is  obtained  by  subtract- 
ing the  sum  of  the  two  given  angles  from  180°.  Then  either  unknown  side  can  b« 
•btained  by  Theorem  I. 

Calling  the  given  side  a,  we  have  b  =  a.  ^!°'      ;  and  cs=a  ^'°" 


sin.  A 


sin.  A 


25 


3S6  TRIGONOMETRY.  [app.  k 

Case  2. — Given  two  sides  and  an  angle  opposite  one  of  them.  The  angle  oppo- 
site the  other  given  side  is  found  by  Theorem  I.  The  third  angle  is  obtained  by 
Bubtracting  the  sum  of  the  other  two  from  180°.  The  remaining  side  is  then  ob- 
tained by  Theorem  L 

Calling  the  given  sides  a  and  b,  and  the  given  angle  A,  we  have  sin.  B  =  sin.  A.  - 

Since  an  angle  and  its  supplement  have  the  same  sine,  the  result  is  ambiguous  ; 
for  the  angle  B  may  have  either  of  the  two  supplementary  values  indicated  by 
the  sine,  if  6  >  a,  and  A  is  an  acute  angle. 

C  =  180°  — (A  +  B).  c  =  sin.G 


sin.  A 


Case  3. — Given  two  sides  and  their  included  angle.  Applying  Theorem  II.  (ob- 
taining the  sum  of  the  angles  opposite  the  given  sides  by  subtracting  the  given 
included  angle  from  180°),  we  obtain  the  difference  of  the  unknown  angles.  Add- 
ing this  to  their  sum  we  obtain  the  greater  angle,  and  subtracting  it  from  their 
eum  we  get  the  less.     Then  Theorem  L  will  give  the  remaining  side. 

Calling  the  given  sides  a  and  b,  and  the  included  angle  C,  we  have 
A-KB  =  180°  — C.    Then 

tan.  i  (A  —  B)  =  tan.  i  (A  +  B) .  ^^^. 
HA-fB)-t.i(A-B)  =  A        i(A-fB)-i(A-B)  =  B.        <'  =  «^|^- 

In  the  first  equation  cot.  |  C  may  be  used  in  the  place  of  tan.  -J  (A  +  B). 

Case .4. — Given  the  three  sides.  Let  s  represent  half  the  sum  of  the  three  sides 
=  -J^  (a  +  6  +  c).  Then  any  angle,  as  A,  may  be  obtained  from  either  of  the  fol- 
lowing formulas,  founded  on  Theorem  III. : 


em.  A  = -^ —^ -" 

be 

J^-fc"  — a* 
cos.  A  ^ 


2  be 

The  first  formula  should  be  used  when  A  <  90°,  and  the  second  when  A  >  90°. 
The  third  should  not  be  used  when  A  is  nearly  180°  ;  nor  the  fourth  when  A  is 
nearly  90°  ;  nor  the  fifth  when  A  is  very  small.  The  third  is  the  most  convenient 
when  all  the  angles  are  required. 


APPENDIX    B. 


DEMONSTRATIONS  OF  PROBLEMS,  ETC. 

Many  of  the  problems,  Ac,  contained  in  the  preceding  pages,  require  Demonstra- 
tions. These  will  be  given  here,  and  -will  be  designated  by  the  same  numbers  ai 
those  of  the  Articles  to  which  they  refer. 

As  many  of  these  Demonstrations  involve  the  beautiful  Theory  of  Transversals, 
«fec.,  which  has  not  yet  found  its  way  into  our  Geometries,  a  condensed  summary 
of  its  principal  Theorems  will  first  be  given. 

TRANSVERSALS. 

Theorem  I. — If  a  straight  line  be  drawn  so  as  to  cut  any  two  sides  of  a  triangle^ 
and  the  third  side  prolonged,  thus  dividing  them  into  six  parts  {the  prolonged  side 
and  its  prolongation  being  two  of  the  parts),  then  will  the  product  of  any  three  of 
those  parts,  whose  extremities  are  not  contiguous,  equal  the  product  of  the  other  three 
parts. 

That  is,  in  Fig.  403,  ABC  being  the  triangle,  and 
DF  the  Transversal,  BEXADXCF=:EAXDCxBF. 

To  prove  this,  from  B  draw  BG,  parallel  to  CA. 
From  the  similar  triangles  BEG  and  AED,  we  have 
BG  :  BE  : :  AD  :  AE.  From  the  similar  triangles 
BFG  and  CFD,  we  have  CD  :  CF  : :  BG  :  BF. 
Multiplying  these  proportions  together,  we  have 
BGXCD:BEXCF::  ADXBG:  AEXBF.  Multi- 
plying extremes  and  means,  and  suppressing  the  common  factor  BG,  we  have 
BEXADXCF  =  EAXDCXBF. 

These  six  parts  are  sometimes  said  to  be  iti  involution. 

If  the  Transversal  passes  entirely  out- 
side of  the  triangle,  and  cuts  the  prolonga- 
tions of  all  three  sides,  as  in  Fig.  404,  the 
theorem  still  holds  good.  The  same  dem- 
onstration applies  without  any  change.* 

Theorem  II. — Conversely :  If  three  points 
be  taken  on  two  sides  of  a  triangle,  and  on 
the  third  side  prolonged,  or  on  the  prolon- 
gations of  the  three  sides,  dividing  them 
into  six  parts,  stich  that  the  product  of 
three  non-consecutive  parts  equals  the  prod- 
uct of  the  other  three  parts;  then  will  these  three  points  lie  in  the  sa)ne  straight  line. 

This  Theorem  is  proved  by  a  Reductio  ad  absurdum. 


*  This  Theorem  may  be  extended  to  polygons. 


388 


TRANSVERSALS. 


[app.  b. 


Theoeem  III.— If  from  the  summits  of  a  triangle,  lines  *'i«-  *56. 

be  drawn,  to  a  point  situated  either  within  or  without  the 
triangle,  and  prolonged  to  meet  the  sides  of  the  triangle, 
or  their  prolongations,  thus  dividing  them  into  six  parts  ; 
then  will  the  product  of  any  three  non-consecutive  parts  be 
equal  to  the  product  of  the  other  three  parts. 

That  is,  in  Fig.  405,  or  Fig.  406, 

AE  X  BF  X  CD  =  EB  X  FC  X  DA. 

For,  the  triangle  ABF  being  cut  by 
the  transversal  EC,  gives  the  relation 
(Theorem  I.), 

AE  X  BC  X  FP  =  EB  X  FC  X  PA. 

The  triangle  ACF,  being  cut  by  the 
transversal  DB,  gives 

DC  X  FB  X  PA  =  AD  X  CB  X  FP. 

,       .  .  P 

Multiplying  these  equations  together, 

and    suppressing  the    common    factors 

PA,  CB,  and  FP,  we  have  AE  X  BF  X  CD  =  EB  X  FC  X  DA. 

Theorem  IV. — Conversely :  If  three  points  are  situated  on  the  three  sides  of  a  tri- 
angle, or  on  their  prolongations  {either  one,  or  three,  of  thise  points  being  on  the  sides) 
so  that  they  divide  these  lines  in  such  a  vjay  that  the  product  of  any  three  non-con 
secutive  parts  equals  the  product  of  the  other  three  parts,  then  will  lines  drawn  from 
these  points  to  the  opposite  angles  meet  in  the  same  point. 

This  Theorem  can  be  demonstrated  by  a  Reductio  ad  absurdum, 

COROLLARIES  OF  THE  PRECEDIXa  THEOREMS. 


CoE.  1. — The  MEDIATES  of  a  triangle  (L  e.,  the  lines  drawn  from  its  summits  to 
the  middles  of  the  opposite  sides)  tneet  in  the  same  point. 

For,  supposing,  in  Fig.  405,  the  points  D,  E,  and  F  to  be  the  middles  of  the  sides, 
the  products  of  the  non-consecutive  parts  will  be  equal,  i.  e.,  AE  X  BF  X  CD  =3 
DA  X  EB  X  FC ;  since  AE  =  EB,  BF  =  FC,  CD  =  DA.    Then  Theorem  IV.  applies. 

CoR.  2.— The  BISSECTRICES  of  a  triangle  (i.  e.,  the  lines  bisecting  its  angles) 
meet  in  the  saine  point. 

For,  in  Fig.  405,  supposing  the  lines  AF,  BD,  CE  to  be  Bissectrices,  we  have 
JLegendre  IV.  17) : 

BF  :  FC  : :  AB  :  AC,  ^  r  BF  X  AC  =  FC  X  AB, 

CD:  DA: :  BC  :  BA,  \  whence  )  CD  X  BA  =  DA  X  BC, 
AE:  EB::  CA:  CB,  )  (  AE  X  CB  =  EB  X  CA. 

Multiplying  these  equations  together,  and  omitting  the  common  factors,  we  hav* 
BF  X  CD  X  AE  =  FC  X  DA  X  EB.     Then  Theorem  IV.  applies. 


App.  b]  transversals.  389 

CoE.  3. — The  ALTITUDES  of  a  triangle  (i.  e.,  the  linjs  arawn  from  its  summita 
perpendicular  to  the  opposite  sides)  meet  in  the  same  point. 

For,  in  Fig.  405,  supposing  the  lines  AF,  BD,  and  CE,  to  be  Altitudes,  we  have 
three  pairs  of  similar  triangles,  BCD  and  FCA,  CAE  and  DAB,  ABF  and  EBC,  by 
comparing  which  we  obtain  relations  from  which  it  is  easy  to  deduce  BF  X  CD  X  AE 
esEBxFCXDA;  and  then  Theorem  IV.  again  applies. 

Cor.  4. — If,  in  Fig.  405,  or  Fig.  406,  the  point  F  be  taken  in  the  middle  of  BC, 
\hen  will  the  line  ED  be  parallel  to  B'C. 

For,  since  BF  =FC,the  equation  of  Theorem  III.  reduces  to  AEXCD=EBX  DA; 
wticnce  AE  :  EB  : :  AD  :  DC  •  consequently  ED  is  parallel  to  BC. 

CoR.  6. — Conversely  :  If 'EL  be  parallel  to  BC,  then  is  BF  =  FC. 

For,  since  AE  :  EB  : :  AD  :  DC,  we  have  AE  X  DC  =  EB  X  AD  ;  whence,  in  the 
jquation  of  Theorem  III.,  we  must  have  BF  :=  FC. 

Cor.  6. — From  the  preceding  Corollary,  we  derive  the  following  : 

If  two  sides  of  a  triangle  are  divided  proportionally, 
parting  from  the  same  summit,  as  A,  and  lines  are  drawn 
from  the  extremities  of  the  third  side  to  the  points  of  divi- 
sion, the  intersections  of  the  corresponding  lines  will  all  lie 
in  the  same  straight  line  Joining  the  summit  A,  ayid  the 
middle  of  the  base. 

CoR.  ^. — A  particular  ease  of  the  preceding  corollary 
is  this : 

In  any  trapezoid,  the  straight  line  which  joins  the  inter- 
section of  the  diagonals  and  the  point  of  meeting  of  the  non-parallel  sides  produced, 
passes  through  the  middle  of  the  two  parallel  bases, 

CoR.  8. — If  the  three  lines  drawn  through  the  corresponding  summits  of  two  trian- 
gles cut  each  other  in  the  same  point,  then  the  three  points  in  which  the  corresponding 
sides,  produced  if  necessary,  will  meet,  are  situated  in  the  same  straight  line. 

This  corollary  may  be  otherwise  enunciated,  thus : 

If  two  triangles  have  their  summits  situated,  two  and  two,  on  three  lines  U'hich 
meet  in  the  same  point,  then,  <tc. 

This  is  proved  by  obtaining  by  Theorem  I.  three  equations,  which,  being  multi- 
plied together,  and  the  six  common  factors  cancelled,  give  an  equation  to  which 
Theorem  II.  applies. 

Triangles  thus  situated  are  called  homologic ;  the  common  point  of  meeting  of 
the  lines  passing  through  their  summits  is  called  the  centre  of  homology  ;  and  the 
hne  on  which  the  sides  meet,  the  axis  of  homology. 


390 


OARMOIVIC  DinSI03l. 


[app.  b 


HARMONIC   DIVISION. 

Dekinitions. — A  straight  line,  AB,  is  said  to  ^'ig-  WS. 

be  harmonically  divided  at  the  points  C  and  D,    I 1 1 —\ 

when  these  points  determine  two  additive  seg-  C        B  D 

ments,  AC,  BC,  and  two  subtractive  segments,  AD,  BD,  proportional  to  one  an- 
other ;  so  that  AC  :  BC  : .  AD  :  BD.  It  will  be  seen  that  AC  must  be  more  than 
Be,  since  AD  is  more  than  BD.* 

This  relation  may  be  otherwise  expressed,  thus :  the  product  of  the  whole  line 
bj  the  middle  part  equals  the  product  of  the  extreme  parts. 

Reciprocally,  the  line  DC  is  harmonically  divided  at  the  points  B  and  A ;  since 
the  preceding  proportion  may  be  written  DB  :  CB  : :  DA  :  CA. 

The  four  points.  A,  B,  C,  D,  are  called  harmonics.  The  points  C  and  D  are  called 
harmonic  conjugates.     So  are  the  points  A  and  B. 

"When  a  straight  line,  as  AB,  is  divided  harmonically,  its  half  is  a  mean  propor- 
tional between  the  distance  from  the  middle  of  the  line  to  the  two  points,  C  and  D, 
which  divide  it  harmonically. 

If,  from  any  point,  O,  lines  be  drawn  so  as  to  Fig.  409. 

divide  a  line  harmonically,  these  lines  are  called 
an  harmonic  pencil.  The  four  lines  which  com- 
pose it,  OA,  OC,  OB,  OD,  in  the  figure,  are 
called  its  radii,  and  the  pairs  which  pass  through 

the  conjugate  points  are  called  conjugate  radii. 

A  C       B  D 

Theorem  V. — In  any  harmonic  pencil,  a  line  drawn  parallel  to  any  one  qf  th« 
radii,  is  divided  by  the  three  other  radii  into  two  equal  parts. 

Let  EF  be  the  line,  drawn  parallel  to  Fig.  410. 

OA.     Through  B  draw  GH,  also  parallel 
to  OA.     We  have, 

GB  :  OA  : :  BD  :  AD  ;  and 
BH :  OA  : :  BC  :  AC. 


But,  by  hypothesis,  AC  :  BC  : :  AD  :  BD. 
Hence,  the  first  two  proportions  reduce  to 
GB  =  BH ;  and  consequently,  EK  =  KF. 

The  Reciprocal  is  also  true  ;  i.  e., 

If  four  lines  radiating  from  a  point  are  such  that  a  line  dravm  parallel  to  one  of 
them  is  divided  into  two  equal  parts  by  the  other  three,  the  four  lines  form  an  har- 
monic pencil. 

*  Three  numbers,  m,  n,  p,  arranged  In  decreasing  order  of  size,  form  an  harmonic  proportioiK, 
when  tlie  difference  of  the  first  and  the  second  is  to  the  difference  of  the  second  and  the  third,  ai 
the  first  is  to  the  third.  Such  are  the  numbers  6,  4,  and  8 ;  or  6,  8,  and  2  ;  or  15, 12,  and  10 ;  &c. 
Bo,  in  Fig.  408,  are  the  lines  AD,  AB,  and  AC,  -which  thus  give  BD  :  CB  : :  AD  :  AC ;  ol 
AC:  CB  ::  AD  :  BD.  The  series  of  fractions,  y,  i,  ^,  1,  -|,  &c.,  is  called  an  harmonic  prog  re«- 
%ion,  because  any  consecutive  three  of  its  terms  form  an  harmonic  proportion. 


B.] 


THE  COMPLETE  QUADRILATERAL. 


391 


Theorem  "VI. — If  any  transversal  to  an  harmonic  pencil  he  drawn,  it  mU  be  divided 
harmonically. 

Let  LM  be  the  transversal.  Through  K,  where  LM  intersects  OB,  draw  EF 
parallel  to  OA.  It  is  bisected  at  K.by  the  preceding  theorem;  and  the  similar 
triangles,  FMK  and  LMO,  EKN  and  LXO,  give  the  proportions 

LM  :  KM  : :  OL :  FK,  and  LN  :  NK  : :  OL  :  EK ;  whence, 
since  FK  =  EK,  we  have  LN  :  NK  : :  LM  :  KM. 

Corollary. — The  two  sides  of  any  angle,  together  with  the  bissectrices  of  the  cungU 
and  of  its  supplement,  form  an  harmonic  pencil. 

Theorem  VII. — If,  from  the  summits  of  any 
triangle,  ABC,  through  any  point,  P,  there  be 
drawn  the  transversals  AD,  BE,  CF,  OTid  the  trans- 
versal ED  be  draicn  to  meet  AB  prolonged,  in  F', 
the  points  F  and  F'  will  divide  the  base  AB  har- 
monically. 

This  may  be  otherwise  expressed,  thus : 

The  line,  CP,  which  joins  the  intersection  of  the  diagonals  of  any  quadrilateral, 
ABDE,  with  the  point  of  meeting,  C,  of  two  opposite  sides  prolonged,  cute  the  side 
AB  in  a  point  F,  which  is  the  harmonic  conjugate  of  the  point  of  meeting,  F',  of 
the  other  two  sides,  ED  and  AB,  prolonged. 

For,  by  Theorem  I.,  AF'  X  BD  X  CE  =  F'B  X  DC  X  EA  ;  and 
by  Theorem  IK,  AF  X  BD  X  CE  =  FB  X  DC  X  EA ; 
whence  AF  :  FB  : :  AF'  :  F'B.  «' 


THE  COMPLETE   QUADRILATERAL. 

A   Complete   Quadrilateral  is  formed  by  Fig.  412. 

drawing  any  four  straight  lines,  so  that  each 
of  them  shall  cut  each  of  the  other  three,  so 
as  to  give  six  different  points  of  intersection. 
It  is  so  called  because  in  the  figure  thus 
formed  are  found  three  quadrilaterals  ;  viz., 
m  Fig.  412,  ABCD,  a  common  convex  quadri- 
lateral ;  EAFC,  a  uni-eoncave  quadrilateral ; 
and  EBAFD,  a  bi-eoncave  quadrilateral,  com- 
posed of  two  opposite  triangles. 

The  complete  quadrilateral,  AEBCDF,  has 
three  diagonals ;  viz.,  two  interior,  AC,  BD ; 
and  one  exterior,  EF. 

Theorem  VIII. — Li  every  complete  quadrilateral  the  middle  points  of  its  three 
diagonals  lie  in  the  same  straight  line. 

AEBCDF  is  the  quadrilateral,  and  LMN  the  middle  points  of  its  three  diago- 
nals.     From  A  and  D  draw  parallels  to  BC,  and  from  B  and  C  draw  parallels  to 


392 


THE  COMPLETE  QUADRILATERAL.  [app.  b. 


Fig.  413. 


AD.  Tl\e  triangle  EDO  being  cut  oy  the  transversal  BF,  we  have  (Theorem  I.), 
DF  X  CB  X  EA  =  CF  X  EB  X  DA.  From  the  equality  of  parallels  between 
paraUels,  we  have  CB  =  E'B',  EA  =  CA',  EB  =  DB',  DA  =  E'A'.  Hence,  the 
ttDove  equation  becomes  DF  X  E'B'  X  CA'  =  CF  X  DB'  X  E'A' ;  therefore,  by 
Theorem  II.,  the  points,  F,  B',  A',  lie  in  the  same  straight  line.  Xow,  since  th« 
diagonals  of  the  parallelogram  ECA'A  bisect  each  other  at  N,  and  those  of  the  par- 
allelogram EBB'D  at  M,  we  have  EN  :  NA' : :  EM  :  MB'.  Then  MN  is  parallel  to 
FA' ;  and  we  have  EN" :  NA'  : :  EL  :  LF,  or  EL  =  LF,  so  that  L  is  the  middle  of 
EF,  and  the  same  straight  line  passes  through  L,  M,  and  N. 

Theorem  IX. — In  every  complete  quadrilateral  each  of  the  three  diagonals  is 
divided  harmonically  by  the  two  others. 

CEBADF  is  the  complete  quadrilateral 
The  diagonal  EF  is  divided  harmonically  at 
G  and  H  by  DB  and  AC  produced ;  since 
AH,  DE,  and  FB  are  three  transversals 
drawn  from  the  supimits  of  the  triangle 
AEF  through  the  same  point  C ;  and  there- 
fore, by  Theorem  VIL,  DBG  and  ACH  di- 
vide EF  harmonically. 

So  to«,  in  the  triangle  ABD,  CB,  CA,  CD, 
are  the  three  transversals  passing  through  C  ;  and  G  and  K  therefore  divide  the 
diagonal  BD  harmonically. 

So  too,  in  the  triangle,  ABC,  DA,  DB,  DC  are  the  transversals,  and  H  and  K 
the  points  which  divide  the  diagonal  AC  harmonically. 

Theorem  X. — If  from  a  point,  A,  any  num-  Fig.  414. 

ber  of  lines  be  drawn,  cutting  the  sides  of  an 
angle  POQ,  the  intersections  of  the  diagonals 
of  the  quadrilaterals  thus  formed  will  all  lie 
in  the  same  straight  line  passing  through  the 
summit  of  the  angle. 

By  the  preceding  Theorem,  the  diagonal 
BC  of  the  complete  quadrilateral,  BAB'C'CO,  ^  ^  ^'  ^"  ^ 

is  divided  harmonically  at  D  and  E.  Hence,  OA,  OP,  OD,  and  OQ,  form  an  har- 
uwnie  pencil  So  do  OA,  OP,  OD',  and  OQ.  Therefore,  the  lines  OD,  OD'  coin- 
cide.    So  for  the  other  intersections. 

If  the  point  A  moves  on  OA,  the  line  OD  is  not  displaced.  If,  on  the  contrary, 
OA  is  displaced,  OD  turns  around  the  point  O.  Hence,  the  point  A  is  said  to  be  a 
pole  with  respect  to  the  line  OD,  which  is  itself  called  the  polar  of  the  point  A. 
Similarly,  D  is  a  pole  of  OA,  which  is  the  polar  of  D.  OD  is  likewise  the  polar  of 
any  other  point  on  the  line  OA ;  and  this  property  is  necessarily  reciprocal  for  the 
two  conjugate  radii  OA,  OD,  with  respect  to  the  lines  OP,  OQ,  which  are  also 
conjugate  radii  Hence :  In  every  harmonic  pencil,  each  of  the  radii  is  a  polar 
with  respect  to  each^oint  of  its  conjugate ;  and  each  point  of  this  latter  line  is  a 
pole  with  respect  to  the  formei. 


DEMONSTRATIONS.* 

PAET  11. ;  CHAPTER  V. 

(140),  (141)  The  equality  of  the  triangles  formed  m  these  methods  proves 
iheir  correctness. 

(143),  (144)  These  methods  depend  on  the  principle  of  the  square  of  the 
hypothenuse. 

(145)  CAD  is  an  angle  Inscribed  in  a  semicircle. 

(146)  Let  fall  a  perpendicular  from  B  to  AC,  meeting  it  at  a  point  E,  not 
marked  in  Fig.  91.     Then  (Legendre,  IV.  12), 

AB'  =  AC  +  BC  —  2  AC .  CE  ;  whence  CE  =         ^  ^^   ~  ^^  . 

2  AC 

BC* 
When  AC  =  AB,  this  becomes  CE  =  — — p,.  The  similar  triangles,  BCE  and  DCA, 

2  AO 

give  EC    CB  : :  AC  :  CD ;  whence 

CE  2  AC         BC 

(14T)  Mark  a  point,  G,  in  the  middle  of  DF,  and  join  GA.     The  triangle  AGD 

■will  then  be  isosceles,  since  it  is  equal  to  the  isosceles  triangle  ABC,  having  two  sides 

and  the  included  angle  equal    Then  AG  =  GD  =  AB  =  GF.    The  triangle  AGF  is 

then  also  isosceles.    Now  the  angle  FAG  =  i  AGD  ;  and  GAD  =  ^  FGA.  Therefore 

FAG  +  GAD  =  FAD  =  i  (AGD  +  FGA)  =  i  (180°)  =  90°. 

(149)  See  Part  VII.,  Art.  (403). 

(150)  The  proof  follows  from  the  equal  triangles  formed. 

(151)  The  proof  is  found  in  the  first  half  of  the  proof  of  Art.  (146). 

(153)  ACP  is  an  angle  inscribed  in  a  semicircle. 

(154)  Draw  from  C  a  perpendicular  to  the  given  line,  meeting  it  at  a  point  R 

AC 
As  in  the  proof  of  Art.  (146),  changing  the  letters  suitably,  we  have  AE  = — —r-. 

The  similar  triangles  AEC  and  ADP  give 

.  ^  .  T,   .  T,  A  T^   ^P   AT.   A.P   AC   AP  X  AG 
AC  :  AE  : :  AP  :  AD  =  ■—  X  AE  =  — -  X  -— ^=  ^  ,„    . 
AC        AC   2  AB    2  AB 

(155)  Similar  triangles  prove  this. 

(156)  The  equal  triangles  which  are  formed  give  BP  ^  CF.     Hence  FP  ia 
parallel  to  BC,  and  consequently  perpendicular  to  the  given  line  DG. 

(157)  The  proof  of  this  is  found  in  the  "  Theory  of  Transversals,"  eor-^llary  S. 
(15§)  The  proof  of  this  is  the  same  as  the  last. 

(161)  The  lines  are  parallel  because  of  the  equal  angles  formed. 

•  Additional  lines  to  the  figures  in  the  text  will  sometimes  be  employed.    The  student  sbonlj 
Irsw  them  on  the  figures,  as  directed. 


394  DEMOXSTRATIOXS  [a^pp.  b. 

(162)  The  equal  triangles  give  equal  angles,  and  therefore  parallels. 

(163)  AB  is  parallel  to  PF,  since  it  cuts  the  sides  of  the  triangle  proportionally. 

(164)  The  proof  is  found  in  corollary  4  of  "  Transversals." 

(165)  From  the  similar  triangles,  CAD  and  CEP,  we  have  CE  :  CD  : :  CP  :  CA. 
From  the  similar  triangles,  CEF  and  CBD,  we  have  CE  :  CD  : :  CF  :  CB.  These 
two  proportions  give  the  following ;  CP  :  CA  : :  CF  :  CB.  Therefore  PF  is  par- 
allel to  AB. 

(166)  Draw  PR  The  similar  triangles  PCE  and  ACD  give  PE :  CE : :  AD :  CD. 
The  similar  triangles  CEF  and  CDB  give  EF  :  CE  : :  DB  :  CD.  These  proportions 
produce  PE  :  EF  : :  AD  :  DB.  Hence  PEF  is  similar  to  ADB,  and  PF  is  parallel 
to  AB. 

(173)  Tlie  equality  of  the  symmetrical  triangles  which  are  formed,  proves  this 
method. 

(174)  ABP  is  a  transversal  to  the  triangle  CDE.  Then,  by  Theorem  L  of 
•'  Transversals,"  CA  X  EB  X  DP  =  AE  X  BD  X  CP ;  whence  we  have 

CP  :  DP  : :  CA  X  EB  :  AE  X  BD. 
By  "division,"  CP  — DP:  DP  ::  CA  X  EB  — AE  X  BD  :  AE  X  BD. 

DC  X  AE  X  BD 


Hence,  since  CP  —  DP  =  CD,  we  obtain  DP : 


CA  X  EB  — AE  XBD 


The  other  formulas  are  simplified  by  the  common  factors  obtained  by  making 
AE  =  AC,  orBE  =  BD.  . 

(175)  By  Theorem  VII.  "  Harmonic  Division,"  in  the  quadrilateral  ABED,  the 
line  CF  cuts  DE  in  a  point,  L,  which  is  the  harmonic  conjugate  of  the  point  at 
which  AB  and  DE,  produced,  would  meet.  So  too,  in  the  quadrilateral  DEHK, 
this  same  line,  CG,  produced,  cuts  DE  in  a  point,  L,  which  is  the  harmonic  conju- 
gate of  the  point  at  which  DE  and  KH,  produced,  would  meet.  Consequently, 
AB,  DE,  and  KH  must  meet  in  the  same  point.  Otherwise;  this  problem  may  be 
regarded  as  the  converse  of  Theorem  X.  of  "  Transversals,"  BCA  being  the  angle, 
and  P  the  point  from  which  the  radiating  lines  are  drawn. 

(176)  EGCFDHisthe  "Complete  Quadrilateral"  Its  three  diagonals  are  FE, 
DC,  and  HG ;  and  their  middle  points  A,  B,  and  P  lie  in  the  same  straight  hne,  bj 
our  Theorem  VIII. 

(1§2)  This  instrument  depends  on  the  optical  principle  of  the  equality  of  the 
angles  of  incidence  and  reflection. 

(184)  The  first  method  given,  Fig.  120,  is  another  application  of  the  Theory  of 
Transversals.  The  second  method  in  the  article  is  proved  by  supposing  the  figure 
to  be  constructed,  in  which  case  we  should  have  a  triangle  QZR,  whose  base,  QR, 
and  a  parallel  to  it,  BD,  wou^d  be  cut  proportionally  by  the  required  line   PSZ  ; 

80  that  QR  :  BD  : :  QP :  BS  =  ^^  ^  ^^ 

QR 

(1§9)  By  "Transversals,"  Theorem  I.,  we  obtain,  regarding  CD  as  the  trans- 
versal of  the  triangle  ABE,  CBX  AFxED  =  ACXFEXDB;  and  since  ED  =  DB. 
this  becomes  CB  X  AF  =  AC  X  FE ;  whence  the  proportion  CB  :  AC  : :  FE  :  AF. 
By   "  division,"    we   have    CB  —  AC  :  AC  : :  FE  —  AF  :  AF.      Observing    that 

AC 
CB  —  AC  =  AB,  we  obtain  AB  =  —  .  (FE  —  AF). 


A.rp.  B.] 


For  Part  II.,  Chapter  T. 


sot 


(190)  Take  CH  =  CB  ;  and  from  B  let  fall  a  perpen- 
dicular, BK,  to  AC.  Then,  in  the  triangle  CBH,  we  have 
JLegendre  IV.  12), 


THg.  124,  bte. 


HK  = 


CH'  +  BH"  — BC      BH" 


2  CH  2  BC ' 

since  CH  =  BC. 

In  the  triangle  ABH,  we  have  (Leg.  IV.  13) 

AB'  =  AH»  4-  BH^  +  2  AH .  HK. 

Substituting  for  HK,  its  value  from  [1],  we  get 

AB»  =  AH«+BH^(l  +  -|5.y 

But  AH  =  AC  -  CH  =  AC  —  BC 

AC  — BC> 


[1] 


AB"  =  AH"  +  BH' 


0+^22)  = 


AH=+BH*. 


AC 


[21 


BC      /  '  —  "BC 

In  the  above  expression  for  AB,  BH  is  unknown.     To  find  it,  proceed  thus. 
Take  CF  =  CD.    Then  DF  is  paraUel  to  BH ;  and  we  have  CD  :  CB  : :  DF  :  BH  ; 

"■"°'  l'H=  =  DF-.^,  M 

In  this  equation  DP  is  unknown ;  but  by  proceeding  as  at  the  beginning  of  this 

CE 
investigation,  we  get  an  equation  analogous  to  [2],  giving  ED'sssEF"  +  DF'  .  -^; 

CD 

whence  DF' =  (DE' —  EF=) .  — . 


Substituting  this  value  of  DF"  in  [3],  we  have 
BH'  =  (DE'— EF«)- 


CB' 


'CDXCE 
Substituting  this  value  of  BH"  in  [2],  we  have 

AB==AH"+(DE"-EF") .  ^g^  =  (AC-BC)'+[DE"-  (CE_CD)"]X^g^. 

(191)  Since  BCD  is  a  right  angle,  AC  is  a  mean  proportional  between  AB 
and  AD. 

(192;   The  proof  follows  from  the  similar  triangles  constructed. 

(193)  The  similar  triangles  give  DE  :  AC  : :  DB  :  AB  ;  whence,  by  "division," 
DE  — AC  :  AC  ::  DB  — AB  :  AB;   whence,  since  DB  — AB  =  AD,   we   have 
ACXAD 


AJ8  = 


DE-AC' 


(194)  From   the   similar  triangles,  we  have   DE  :   CA::EB  :  AB ;  whence 
DE  — CA  :  CA  ::  EB  — AB  :  AB ;    whence,    since    EB  — ABsrAE,  we    get 
AC  X  AE 


AB  = 


DE  —  AC' 


(195)  The  triangles  DEF  and  BAF,  similar  because  of  the  parallelogram  which 
j9  constructed,  give  FE  :  ED  : :  AF  :  AB  :=  — ^tft—  =  — ff — • 

ACXDC 
The  triangles  DEF  and  BCD  give  similarly  FE  :  ED  : :  DC  :  CB  =  -^^g — . 


396  DEi^lOXSTRATIOXS  [app-  b. 

(196)  The  equality  of  the  triangles  formed  proves  this  problem. 

(197)  The  proof  of  this  problem  also  depends  on  the  equality  of  the  triangle* 
eonstructed.     The  details  of  the  proof  require  attention. 

(198)  EB  is  the  transversal  of  the  triangle  ACD.     Consequently,  CBxAFxDE 
=  ABXFDXCE;  or,  sineeCB  =  AB+AC,  (AB+AC)XArxDE=ABxFDxC^  ; 

ACXAFXDE 


whence  AB  ^ 


FDXCE  — AFXDE" 


Taking  E,  in  the  middle  of  CD,  CE  =  DE,  ana  those  lines  are  cancelled.  Tabling 
F  in  the  middle  of  AD,  AF  ^  FD,  and  those  hnes  are  cancelled. 

(199)  The  line  BE  is  harmonically  divided  at  the  points  H  and  A,  from  The  >rem 

IX.,  ECFBGD  being  a  "  Complete  Quadrilateral."  Consequently,  AE :  EH : :  AB :  HB. 

Hence,  by  "division,"  AE  — EH  :  AE  : :  AB  —  HB  :  AB.     We  therefore  have, 

A P  y  ATT 
since  AB  —  HB  =  AH,  AB  =  ,„\,^- 

(200)  For  the  same  reasons  as  in  the  last  article,  CF  is  harmonically  divided  at  H 

and  D  ;  and  we  have  CH :  HF  : :  CD  :  DF ;  whence  CH  — HF :  CH : :  CD  — DF :  CD 

OTT  y  OP 

Hence,  since  CD  —  DF  =  CF,  CD  =  — -— . 

Cx±  —  i±r 

The  other  two  expressions  come  from  writing  CF  as  CH  +  HF,  and  HF  as 
CF  —  CH. 

(201)  The  equality  of  the  triangles  formed  proves  the  equality  of  the  corre- 
Bponding  sides  KD  and  DE,  (fee. 

(202)  The  similar  triangles  (made  so  by  the  measurement  of  CE)  give 
CD:DE::CA:AB  =  ^^^. 

(203)  The  similar  triangles  (made  so  by  the  parallel)  give  CE  :  EA  : :  CD  :  A.B 
CDXEA_CDX(AC  — CE) 

~       CE  CE 

DFxCD 

(204)  The  similar  triangles  DFH  and  BCD  give  HF  :  FD  : :  DC  :  BC  =      ^^ 

CH 

The  similar  triangles  FGH  and  ABC  give  FG  :  GH  : :  BC  :  AB  =BC  ^. 

"DF  y  CD  V  C  H 

Substituting  for  BC,  its  above  value,  we  have  AB  =:  — ==^ =-- — . 

r a.  X  -cG 

When  CD  =  CE,  DF  =  CD,  whence  the  second  formula. 

(205)  The  equality  of  the  symmetrical  triangles  which  are  formed,  pioves  the 
<squality  of  A'B'  to  AB. 

(206)  The  proof  of  this  is  similar  to  the  preceding. 

(207)  Because  the  two  triangles  ABC  and  ADE  have  a  common  angle  at  A, 
we  have  ADE  :  ABC  : :  AD  X  AE  :  AB  X  AC ;  whence  the  expression  for  ABC. 

(20§)  From  B  let  fall  a  perpendicular  to  AC,  meeting  it  at  a  point  B'.  Call 
this  perpendicular  BB'  =  p.  From  D  let  fall  a  perpendicular  to  AC,  meeting  it 
»t  a  point  D'.     Call  this  perpendicular  DD'  =  q. 


[app.  3.]  For  Part  V.  39^ 

The  quadrilateral  A13CD  =  AC  X  i  (;?  +  5). 

The  triangle  BCE  =  CE  X  i  p  ;  whence  p  =    '       ■■ . 

Gil 

The    similar   triangles   EDD'    and    BEB'   give  p   :   g   :  :   BE   :   DE,   whencn 

_     DE  _  2  .  BCE  X  DE 

'  ~^  BE  —      CE  X  BE     ■ 

m       W.N      ^^^  I  BCEXDE      ^^^      BE-f-DE      ^^^  BD 

Then  i  (p  +  y)  =  — -  + -T^^S^  =BCE  X  — ^-,==BCE  X 


CE         CEXBE  CEXBE  CExBE' 

Lastly.  ABCD  =  AC  X  BCE  X  ^^^  =  BCE  X  ^g||. 


DEMONSTRATIONS  FOR  PART  V. 

(3§2)  Let  B  =  the  measuTed  inclined  length,  6  =  this  length  reduced  to  a 
horizontal  plane,  and  A  ^  the  angle  which  the  measured  base  makes  with 
the  horizon.  Then  6  :=  B  .  cc«.  £  and  the  excess  of  B  over  b,  L  e., 
B  —  6  =  B  (1  —  COS.  A).  Since  1  —  cos.  A  =  2  (sin.  -§■  A)"  [Trigonometry, 
Art.  (9)],  we  have  B  —  6  =  2  B  (sin.  -J  A)^  Substituting  for  sin.  |.A,  its 
approximate  equivalent,  -J  A  X  sin.  1  [Trigonometry,  Art.  (5)],  we  obtain 
B  — 6  =  2B  (iAX  sin.  I'f  =  i  (sin.  l')^  A'.  B,  =0.00000004231  A"  B 
By  logarithms,  log.  (B—  6)  =  2.626422  4-  2  log.  A  +  log.  B.  The  greater  precision 
of  this  calculation  than  that  of  6  =  B  .  cos.  A,  arises  from  the  slowness  with  which 
the  cosines  of  very  small  angles  increase  or  decrease  in  length. 

(386)  The  exterior  angle  LER  =  LCR  +  CLD.     Also,  LER  =LDR  +  CRD. 
.•.LCR+CLD  =  LDR4-CRD,      and  LCR  =  LDR+CRD-CLD. 

CD 

Fi-om  the  triangle  CRD  we  get  sin.  CRD  :=  sin.  CDR  X  -^. 

CR 

CD 
From  the  triangle  CLD  we  get  sin.  CLD  ^  sin.  LDC  X  -7-. 

CL 

As  the  angles  CRD  and  CLD  are  very  small,  these  values  of  the  sines  may  be 
tailed  the  values  of  the  arcs  which  measure  the  angles,  and  we  shall  have 

CD  cn 

LCR  =  LDR  +  sin.  CDR  X  —  —  sin.  LDC  X  ^. 
Civ  CL 

The  iast  two  terras  are  expressed  in  parts  of  radius,  and  to  have  them  in  seconds, 
they  must  be  divided  by  sin.  i'  ^Trigonometry,  Art.  (5),  Note],  which  gives  the 
formula,  in  the  text.  Otherwise,  the  correction  being  in  parts  of  radius,  may  be 
brought  into  seconds  by  multiplying  it  by  the  length  of  the  radius  in  seconds ;  i.  e., 
1  fio°  y  fin  y  fift 
— J =  206264".80625  [Trigonometry,  Art.  (2)  ]. 

(391)  The  triangles  AOB,  BOC,  COD,  &c.,  give  the  following  proportions 
[Trigonometry,  Art.  (12),  Theorem  I.]  ;  AO  :  OB  :  :  sin.  (2)  :  sin.  (1) ; 
OB  :  OC  : :  sin.  (4) :  sin.  (3) ;  OC  :  OD  : :  sin.  (6) :  sin.  (5) ;  and  so  on  around  the 
polygon.  Multiplying  together  the  corresponding  terms  of  all  the  proportions, 
the  sides  will  all  be  cancelled,  and  there  will  result 

1  :  1  : :  sin.  (2)  X  sin.  (4)  X  sin.  (6)  X  sin.  (8)  X  sin.  (10)  X  sin.  (12)  X  sin.  (14) : 

sia  (1)  X  sin.  (3)  X  sin.  (5)  X  sin.  (7)  X  sin.  (9)  X  sin.  (11)  X  sin.  (18). 
3ence  the  equality  of  the  last  two  terms  of  the  proportion. 


398  DETIOXSTRATIONS 


DEMONSTRATION  FOR  PART  VI, 
(399)  In  the  triangle  ABS,  we  have 

sin.  ASB  sin.  S 


Ao-D     •     r.AC3        AT>    q^  _  AB .  sin.  BAS       c.sin.  U 

sm.  ASB  :  sm.  B  A.S  : :  AB  :  SB  = ; — -— - —  =  — -, .  rn 

Kin    Ask  oin  s«  J 


In  the  triangle  CBS,  we  have 

B( 
"Rr!-SR=_ 

sin.  BSO  sia  S' 


•     T.ci^      •     T>^c.        T->^    c.T>       BC .  sin.  BCS       a.sin.  V  ^  , 

sin,  BSC  :  sin.  BCS  : :  BC  :  SB  = -. — ^— -- —  =  — : — -— .  [21 


„  c.sin.  U       a.8in.V       ,  .     „,     .    ^  .     „     .     „ 

Hence,  — : — —-  =  — : — — -  ;  whence,  c .  sia  S  .  sin.  U—a .  sm. S  .  sin.  V=0.  [81 
sin.  S  sm.  S  ^  -' 

In  the  quadi-ilateral  ABCS,  we  have 
BCS  =  360°  — ASB  — BSC  — ABC  — BAS;  or  V  =  360°— S  — S'  — B  — U. 

LetT  =  360°  — S  — S'— B,  and  wehave  V  =  T  — IT.  [*1 

Substituting  this  value  of  V,  in  equation  [3],  we  get  [Trig.,  Art.  (8)], 

c , sin  S'  sin.  U  —  a.  sin.  S  (sin.  T .  cos.  U  —  cos.  T .  sin,  U)  =  0 

Dividing  by  sin.  U,  we  get 

.     ^,  .     f>  /  .     „,   cos.  U  \ 

c  .  sin.  S  —  a  .  sm.  S  I  sm.  T .  -; — — :  —  cos.  T  I  =  0 
\  sm.  U  / 


cos.  U 
,  sin.  b'  —  a  .  sm.  S  I  sm.  T . 

Whence  we  have 


cos.  U  ,   ^^       c.  sin.  S  4-  a .  sin.  S  .  cos.  T 

—  =  cot.  U  = r 7Z : = . 

sm.  U  a .  sin,  S ,  sin.  T 

Separating  this  expression  into  two  parts,  and  cancelling,  we  get 
,    ^^  c .  sin.  S'         .   cos.  T 

cot,  U  = : : —  + — . 

a  .  sm.  S  ,  sm.  T  ^  sm.  T 

Separating  the  second  member  into  factors,  we  get 

cos.  T  /       c .  sin.  S'  \ 

cot.  U  =  I 1-  1  I  :  or 

sin.  T  Va.sin.  S.cos.  T  ^    /  ' 

c .  sin.  S' 


„  /       c .  sin.  S  .      \ 

cot  U  =  cot.  T  ( ^-- ^  +  1 ) 

\a .  sin.  S .  cos.  T         / 


Having  found  TJ,  equation  [4]  gives  V ;  and  either  [1]  or  [2]  gives  SB ;  and 
SA  and  SC  are  then  given  by  the  familiar  "  Sine  proportion"  [Trig,  Art.  (12)] 


App.  B.]  For  Part  VII.  398 


DEMONSTRATIONS  FOR  PART  VH. 

CP 
.     (403)  If  APC  be  a  right  angle,  —  =  cos.  CAB  [Trigonometry,  Art.  (4)], 

(405)  AC  =  PC .  tan.  APC ;  and  CB  =  PC .  taa  BPC  [Trigonometry,  Art.  (4)]. 
Henc«  AC  :  CB  : :  tan.  APC  :  tan,  BPC ;  and 

AC  :  AC  +  CB  : :  tan.  APC  :  tan.  APC  +  tan.  BPC. 

Consequently,  since  AC  +  CB  =  AB,      AC  =  AB tan.  APC 

^         ■"  tan.  APC -f  tau.  BPC 

(414)  The  equal  and  supplementary  angles  formed  prove  the  operation. 

(421)   In  Fig.  285,  CA  :  EG  : :  AB  :  GB.      By  "  division,"  CA  — EG  :  EG  : : 
AB  —  GB  :  GB.      Hence,    observing    that    AB  —  GB  =  AG,   we    shall    have 
,^_GB(CA-EG) 
^^ EG • 

(423)      Art.    (12),    Theorem    III.,    [Trigonometry,    Appendix    A,]    gives 

^2  J-  J" c' 

cos.  C  = ; ;  or  c'  =  a*  +  6^  —  2  ab  .  cos.  C.      This  becomes  [Trig.,  Art 

2ab  L      &. 

(6)],  K  being  the  supplement  of  C,  c^  =  a' -\-  b'  -\- 2  ab  .  cos.  K.  The  series  [Trig. 
Art.  (5)]  for  the  length  of  a  cosine,  gives,  taking  only  its  first  two  terms,  since  K  ia 
very  small,  cos.  K  =  1  —  i  K'.     Hence, 

c''  =  a»  +  6»  +  2  a6  —  a6  K'  =  (a  +  bf-ab  K^  =  (a  +  J)'  ( 1  —  /'\^')  ; 

\         (a+b)  / 

whence,  e  =  (a+ 6)  ^(l- ^?^,). 

Developing  the  quantity  under  the  radical  sign  by  the  binomial  theorem,  and  neg- 
lecting  the  terms  after  the  second,  it  becomes 

(a  +  6)' 

Substituting  for  K  minutes,  K .  sin.  1'  [Trig.,  Art.  (5)],  and  performing  the  multi* 
plication  by  a  +  6,  we  obtain 

c=a+b-^^^^\     Xow^-!lBli:^  =  0.0000000423079; 
^  2  (a  +  6)  2 

whence  the  formula  in  the  text,  c  =  a  +  6  —  0.000000042308  X  — rr- 

a  -^  0 

(430)  In  the  triangle  ABC,  designate  the  angles  as  A  B,  C;  and  the  sides  op- 
posite to  them  as  a,  b,  c.    Let  CD  =  d.  .  The  triangle  BCD  gives  [Trig,  Art.  (12), 

_  TT  ,  sin.  BDC     ^        .       ,     .  ^^    .    .,    ,      .        ,        ,    sin.  ADC 

Theorem  11,  a  =  d  - — r— — -.     The  triangle  ACD  similarly  gives  o  =  a'  -. — jr-rFi- 

sm.  CBD  °  •'  ^  sm.  CAD 

In  the  triangle  ABC,  we  have  [Trig.,  Art.  (12),  Theorem  II.], 

tan.  ^  (A  —  B)  :  cot.  i  G  ::  a  —  b  :  a-\- b; 

whence  tan.  -^  (A  —  B)  =  ^^  •  cot.  i  C.  [1] 

a  +  b 

b 
Let  E  be  an  auxiliary  angle,  such  that  b  =  a .  tan.  K ;  whence  tan.  K  ^  — 


40U  DEMONSTRATIONS  [app.  b. 

Dividing  the  second  member  of  equation  [1],  above  and  below,  by  a,  and  substitu- 
ting tan.  K  for  -,  we  get  tan.  4  (A  —  B)  =-— ; '—^^  •  cot.  |  C. 

"  a         ^  ^  1  +  tan.  K 

Since  tan.  45°=  1,  we  may  substitute  it  for  1  in  the  preceding  equation,  and. 

1  / «       -D\      tijn.  45°  —  tan,  K 

we  get  tan.  i  (A — B)  = --r—, ^'Cot.  i  G. 

^  ^  ^  ''tan.  45°  +  tan.  K 

From  the  expression  for  the  tangent  of  the  difference  oi  two  arcs  [Trig.,  Art. 
(8)],  the  preceding  fraction  reduces  to  tan.  (45°  —  K) ;  and  the  equation  becomes 

tan.  |(A  —  B)  =  tan.  (45°  —  K) .  cot.  i  C.  [2] 

In  the  equation  tan.  K  =  — ,  substitute  the  values  of  b  and  a  from  the  formulas 
a 

at  the  beginning  of  this  investigatioa     This  gives 

sin.  ADC  .  ,  sin.  EDO   sin.  ADC .  sin.  CBD 


tan.  K  =  d- 


sin.  CAD  •   sin.  CBD   sin.  CAD .  sin.  BDC' 


(A  —  B)  is  then  obtained  by  equation  [2]  ;  (A  +  B)  is  the   supplement  of  C 
therefore  the  angle  A  is  known. 

_        _a.sin.  C_c?.sin.  BDC.sin.  ACB 
^°  '^~  sin.  A  sin.  CBD .  sin.  CAB  ' 

The  use  of  the  auxiliary  angle  K,  avoids  the  calculation  of  the  sides  a  and  b. 

(434)  In  the  figure  on  page  292,  produce  AD  to  some  point  F.     The  exterior 

angles,  EBC  =  A  +  P;    ECD  =  A  +  Q  ;     EDF  =  A  +  R.      The  triangle   ABE 

BE       sin.  A     _,         .       ,     ,    „     .  CE         sin.  A      .^.  . ,. 

gives  —  =  -: — — .     The  triangle  ACE  gives  =  -: — — .      Dividing  member 

^  a        sm.  P  °  ^         a  +  x       em.  Q  * 

,  ,  ,  BE  a. sin.  Q 

by  member,  we  get  -—  = 


CE       (a  -f  x)  sin.  P' 

In   the   same  way  the  triangles   BED  and  CED  give  -r—. —  =   .  '  \^ — - ; 

•'  °  ^        b-\-x      sin.  (R  —  P) 

,  CE       sin.(A  +  R)      ^,  u  r       BE       (6  +  a;)  sin.  (R— Q) 

and  —  =  .     '         ^!.     Whence  as  before,  --  =      T    ■     ,^      ^,      ■ 
b        sm.  (R  —  Q)  CE  6 .  sin.  (R  —  P) 


Equating  these  two  values  of  the  same  ratio,  we  get 
a.  sin.  Q       _ 
(a  -j-  a;)  sin.  P 
ab .  sin.  Q .  sin.  (R  —  P) 


a.sin.  Q  (6  +  a:)  sin.  (R  —  Q) 

■     .     ,   : — 5  =  — 7 — r—7^ ^j; — - ;  and  thence 

a  -j-  a;)  sm.  P  6  .  sin.  (R  —  P) 


sin.  P .  sin.  (R  —  Q)  ^     ■     /  \     >     /  i   \     i     ^ 

To  solve  this  equation  of  the  2d  degree,  with  reference  to  x,  make 

tan »  K  =     ^"^      .  ""-QC^iP-R-P) 
(a  —  bf    sin.  P  (sin.  R  —  Q)' 

Then  the  first  member  of  the  preceding  equation  =  i  •  (a  —  6)'  X  tan.'  K '  and 
we  get  a:''+{a+b)x  =  i{a  —bf  •  tan."  K  —  ab, 

and  z  =  —  i{a  +  b)±^[i  (a  —  6)"  •  tan.' K  —  aJ  +  i  (a  +  6)'! 

=  —h(a  +  b)±y/[i{a  —  6)'. tan."  K+i{a  —  bf] 

—  —  i{a+b)±i{a  —  b)y/  (tan.'  K  +  1). 

Or,  since  y/ (tan.'  K  -f  1)  =  secant  K  =  —^, we  have  x=z  —  ^-^  ±    """^^ 

cos.  K  2  2.  COS.  K 


APP.   B.l 


For  Part  XI. 


401 


DEMONSTRATIONS  FOR  PART  XI. 


(493)  The  oontent  being  given,  and  the  length  to  be  n  times  the  breadth 
Breadth  X  n  times  breadth  =  content ;  whence,  Breadth  =a/  ( ). 

Given  the  content  =  c,  and  the  difference  of  the  length  and  breadth  =  d ;  to 
find  the  length  I,  and  the  breadth  6.  We  have  I  X  b^c  ;  and  I  —  b  =  d.  Frona 
these  two  equations  we  get  I  =  i  d -{-  i  ^  {d^ -\-  4:  c). 

Given  the  content  =  c,  and  the  sum  of  the  length  and  breadth  ^  s ;  to  find  I  and  b. 
We  have  I  X  b  =  c;  and  I  -{•  b^=s;  whence  we  get  i  =  ^  s  -(-  *  \^  (s'  —  4  c). 

(494)  The  first  rule  is  a  consequence  of  the  area  of  a  triangle  being  the  product 
of  its  height  by  half  its  base. 

To  get  the  second  rule,  call  the  height  A ;  then  the  base  =  »«A ;  and  the  area 

,  .           ,        ,           ,           /  /2  X  area\ 
=  ^ rt  X  mfi ;  whence  h  =  A/   I I. 

For  the  equilateral  triangle,  calling  its  side  e,  the  formula  for  the  area  of  a  triangle 
V[iis)ihs  —  a){is  —  b){is  —  c)-]  reduces  to  i  e»  ^3.  Hence  e  =  2  a/ (^\ 
=  1.5197  ^area. 

(495)  By  Art.  (65),  Note,  i .  AB  X  BC  X  sin.  B  =  content  of  ABC  ;  whence, 
2  X  ABO 


BC  = 


AB.sin.  B* 


22  // 

(496)  The  area  of  a  circle  ^radius*  X  -=- ;  whence  radius^  i/  I 


T  X  area\ 


22 


; 


(497)  The  blocks,  including  half  of  the  streets  and  avenues  around  them,  are 
900  X  260  =  234000  square  feet.  This  area  gives  64  lots ;  then  an  acre,  or  43560 
feet,  would  give  not  quite  1 2  lots. 

(502)  The  parallelogram  ABDG  being  double  the  triangle  ABC,  the  proof  for 
Art.  (495),  slightly  modified,  applies  here. 


(504)  Produce  BO  and  AD  to  meet  in  E. 
By  similar  triangles, 

ABE  :  DCE  : :  AB^ :  DC. 
ABE  —  DCE  :  ABE  : :  AB"  —  DC  :  AB'' 

Now  ABE  —  DCE  =  ABCD ;    also,    by 
Art.  (65),  Note, 

ABE  =  AB^^^°•.^•^^^. 
2.sm.  (A+B) 

The  above  proportion  therefore  becomes 

ein.  A .  sin.  B 


Fig.  846,  tis. 


ABCD:  AB». 


AB«-i.CD»:AB». 


2  .  sin.  (A  +  B) 

Multiplying  extremes  and  means,  cancelling,  transposing,  and  extracting  the  eqtlbra 

2  .  ABCD .  sin.  (A  +  B)! 
sin.  A.  sin.  B         J' 
26 


root,  we  get  CD  =  i/  FaB" 


t02  DEMONSTRATIONS  [app.  b 

"When  A  -|-  B  >  180°,  sin.  (A  -f  B)  is  negative,   and  therefore  the  fraction  in 
which  it  occurs  becomes  positive. 
CF  being  drawn  parallel  to  DA,  we  have 

(505)  Since  similar  triangles  are  as  the  squares  of  their  homologous  udeti 
BDE  :  BFG  : :  BD« :  BF^  whence  BF  =  BI>i/(|^). 

(506)  BFG  =  i  .  BF  X  FG  =  i .  BF  X  BF .  tan.  B  ; 

//2.BFG\ 
whence,  ^^  =  f  (i^^Tb)- 

(510)  By  Art.  (65),  Note,  BFG  =  BP .  ij^^jl^|^^ ; 

,  „„         //2.8in.  (B  +  F).BFG\ 

whence,  BF ;—  '  '^  '  v      •      /  i 


■=/(^ 


sin.  B  .  sin.  F         / 

(511)  The  final  formula  results  from  the  proportion 

FAE  :  CDE  : :  AE* :  ED^ 

(512)  Since  triangles  which  have  an  angle  in  each  equal,  are  as  the  produotioi 
the  sides  about  the  equal  angles,  we  have 

ABE  :  CDE  : :  AE  X  BE  :  CE  X  DE. 

ABE  =  i■AB^^^°•f•t•^.     AE  =  AB.i^. 
sm.  (A  -j-  B)  sm.  E 

BE  =  AB.!!^.  CE  =  DE.«i^. 

sm.  E  sm.  DCE 

Substituting  these  values  in  the  preceding  proportion,  cancelling  the  common  fac 
tors,  observing  that  sin.  (A  +  B)  ^  sin.  E,  multiplying  extremes  and  means,  and 
A-  -A-  .  TM7  //2.CDE.8in.  DCE\ 

dxvidmg,  we  get  DE  =  |/ (  3^,.  e  .  sia  CDE  I 

(515)  The  first  formula  is  a  consequence  of  the  expression  for  the  area  of  a 
triangle,  given  in  the  first  paragraph  of  the  Note  to  Art.  (66). 

(517)  The  reasons  for  the  operations  in  this  article  (which  are  of  very  frequent 
occurrence),  are  self-evident. 

(51  §)  The  expression  for  DZ  follows  from  Art.  (65),  Note.  The  proportion  in 
the  next  paragraph  exists  because  triangles  having  the  same  altitude  are  as  their 
basea 

(519)  By  construction,  GPC  =  the  required  content.  Now,  GPC  =  GDC,  since 
they  have  the  same  base  and  equal  altitudes.  We  have  now  to  prove  that 
LMC  ^  GDC.  These  two  triangles  have  a  common  angle  at  C.  Hence,  they  are 
to  each  other  as  the  rectangles  of  the  adjacent  sides;  L  e., 

GDC  :  LMC  : :  GO  X  CD  : :  LC  X  CM. 

Here  CM  is  unknown,  and  must  be  eliminated.  "We  obtain  an  ezpressicm  for  it 
by  means  of  the  similar  triangles  LCM  and  LEP,  which  give 

LE  :  LC  : :  EP  =  CD  :  CM. 


App.  B.]  For  Part  XI,  403 

CD  X  LC 
Hence,  CM=: -— — .    Substituting  this  value  of  CM  in  the  first  proportion, 

and  cancelling  CD  in  the  last  two  terms,  we  get 

GDC  :  LMC  : :  GC  :  -p— ;  or  GDC  :  LMC  : :  GC  X  LE  :  LC». 

LC  =  (LH  +  HCf  =  LH'  +  2  LH  X  HC  +  HC 
But,  by  construction, 

LH»  =  HK' =  HE»- EK=  =  HE'- EC  =(HE+EC)  (HE-EC)  =  HC  (HE-EC)l 
Also,  GC  =  2  HC  ;  and  LE  =  LH  +  HE. 

Substituting  these  values  in  the  last  proportion,  it  becomes 

GDC  :  LMC  : :  2  .  HC  (LH  +  HE) :  HC  (HE  —  EC)  +  2  LH  X  HC  +  HC. 
::2LH  +  2HE   :  HE  — EC  +  2  LH  +  HC. 

:  HE  —  EC  +  2  LH  +  HE  +  EC. 
:  2  HE  +  2  LH. 

The  last  two  terms  of  this  proportion  are  thus  proved  to  be  equal.    Therefore,  the 
first  two  terms  are  also  equal ;  I  e.,  LMC  ^  GDC  =  the  required  content. 

Since  HK  =  y/  (HE'  —  EK'),  it  will  have  a  negative  as  well  as  a  positive  value. 
It  may  therefore  be  set  off  in  the  contrary  direction  from  L,  i.  e.,  to  L'.  The  line 
drawn  from  L'  through  P,  and  meeting  CB  produced  beyond  B,  will  part  off  an- 
other triangle  of  the  required  content. 

(520)  Suppose  the  line  LM  drawn.  Then,  by  Art.  (65),  Note,  the  required 
content,  c  =  i  •  CL  X  CM .  sia  LCM.  This  content  will  also  equal  the  sum  of  thfi 
two  triangles  LCP  and  MCP  ;  i.  e.,  c  =  i  •  CL  X  p  +  i  '  Cii  X  q.     The  first  of 

2  c 

these  equations  gives  CM  = —^: : — Trrrr.-     Substituting  this  in  the  second  equa- 

CL .  sm,  LCM 
tdon,  we  have  ,    „^  .  co 

^  =  ^-^^^^  +  cl7S^lcm- 

Whence,  ip.CL''.  sin.  LCM  +  cq  =  c.CL.  sin,  LCM. 

Transposing  and  dividing  by  the  coefficient  of  CL*,  we  get 

2  c  "  '•'• 

CL» CL; 


p     y  \p^    p 


p  .  sin.  CLM 
2cq 


sin.  LCM  J 


If  the  given  point  is  outside  of  the  lines  CL  and  CM,  conceive  the  desired  line 
to  be  drawn  from  it,  and  another  line  to  join  the  given  point  to  the  corner  of  the 
field.  Then,  as  above,  get  expressions  for  the  two  triangles  thus  formed,  and  put 
their  sum  equal  to  the  expression  for  the  triangle  which  comprehends  them  both, 
and  thence  deduce  the  desired  distance,  nearly  as  above. 

(522)  The  differenced,  between  the  areas  parted  off  by  the  guess  line  AB,  ana 
the  required  line  CD,  is  equal  to  the  difference  between  the  triaiigles  AFC  and  BPD 

By  Art.  (65),  Note,  the  triangle  APC  =  i-AF'-^!°'  f  '  ""' ^. 
•'  '  ^  ^  sin.  (A  -j-  P) 

Bimilarly  the  triangle  BPD  =  4  •  BP«  "°'  ^ '  ^]^'  ^. 
•'  ^  ^  em.  (B  +  P) 

••'*-*^^    8in.(A  +  P)       *^^      8in.(B  +  P^ 


404  DEMONSTRATIONS  [app.  b 

By  the  expression  for  sin.  (a  +  &)  [Trigonometry,  Art.  (8)],  we  have 


sin.  A  .  COS.  P  +  sin.  P .  cos.  A  sin.  B .  cos.  P  +  sin.  P .  cos.  B 

Dividing  each  fraction  by  its  numerator,  and  remembering  that  ^-^  =  cot.  a,  we 
o  J  "^  sin.  a 

have  i  AF' i  BP' 

~  cot.  P  +  cot.  A       cot.  P  +  cot.  B' 

For  convenience,  let  p  =  cot.  P  ;  a  =  cot.  A  ;  and  b  ^  cot.  B.    The  above  equatiot 
will  then  read,  multiplying  both  sides  by  2, 

AP''  BP 

2  d= . 

p  •{■  a,      p-^-b 

Clearing  of  fractions,  we  have 

2  (fjs"  +  2  dap  -{■  2  dbp  +  2  dabz=p  .  AF'  +  b  .  AP"  —p.BV^  —  a.  BP» 
Transposing,  dividing  through  by  2  d,  and  separating  into  factors,  we  get 
AP^  —  BP"  \  6 .  AP"  —  a .  BP2 


.    /      .    I       AP^— BP-''\  I 

P^+{a  +  b rd—)p=- 


2  d 


(lb. 


I              AP=— BPn   ,      /r&.AP'  — a.BP2                 /              AP^  — BP=\2  i 
•••^=-H"  +  ' 2T-)±^4 T5- «5  +  i(«  +  6 25— )  J- 

If  A  =  90°,  cot.  A  =  a  =  0 ;  and  the  expression  reduces  to  the  simpler  form 
given  in  the  article. 

(523)  Conceive  a  perpendicular,  BF,  to  be  let  fall  from  B  to  the  required  line 

DE.     Let  B  represent  the  angle  DBE,  and  /3  the  unknown  angle  DBF.     The  angle 

BDF  =  90°  —  /? ;  and  the  angle  BEF  =  90°  —  (B  —  j3)  =  90°  —  B  +  /3.     By  Art. 

,...    -VT  .        .  .     ,  .  ^^„       ,    ^^       sin.  BDE  .  sin,  BED 

(65),   Note,   the   area   of   the   triangle   DBE  =  \   DE^  •  —. — -^^   ,  .p^^.    = 

sin.  (BDiii  -|-  BiiiiJ) 

J  .  UE= .  sin.(90°-/3)sin.  (90°-B  +  /?) 

sin.  B 

„  ^„,  2  X  DBE  X  sin.  B  2  X  DBE  X  sin.  B 

Hence,     DE=  = =: . 

sm.  (90°  —  /3) .  sin.  (90°  —  B  + 13)        cos.  /? .  cos.  (B  —  j3) 

Now  in  order  that  DE  may  be  the  least  possible,  the  denominator  of  the  last 
fraction  must  be  the  greatest  possible.  It  may  be  transformed,  by  the  formula, 
COS.  a.  cos.  6  =  i  cos.  (a -j- 6) -|- ^  .  cos.  (a  —  b)  [Trigonometry,  Art.  (8)],  into 
\  cos.  B  -f  -^  .  cos.  (B  —  2  (8).  Since  B  is  constant,  the  value  of  this  expression  de- 
pends on  its  second  term,  and  that  will  be  the  greatest  possible  when  B  —  2  /3  :=  0, 
in  which  case  |8  =  ^  B. 

It  hence  appears  that  the  required  line  DE  is  perpendicular  to  the  line,  BF, 
which  bisects  the  given  angle  B.    This  gives  the  direction  in  which  DE  is  to  be  rua 

Its  starting  point,  D  or  E,  is  found  thus.  The  area  of  the  triangle 
DBE  ^  \  BD .  BE  .  sin.  B.     Since  the  triangle  is  isosceles,  this  becomes 

DBE  =  i  BD» .  sin.  B  ;  whence  BD  =  J  C^^^^- 

'\     \  sm.  B  / 

DE  is  obtained  from  the  expression  for  DE",  which  becomes,  making  /^^-^B. 

_^       2  X  DBfi  X  sin.  B       ,  ^^       .^(2  .  DBE .  sin.  B) 

DE=  = —5 —5-  ■  whence,  DE  =  ^^ -^ ■' 

cos,  i  B ,  cos.  \  B  cos,  \  B 


App.  B.]  For  Part  XI.  405 

(524)  Let  a  =  value  per  acre  of  one  portion  of  the  laud,  and  h  that  of  the 
ether  portion.     Let    x  =  the  width  required,  BC  or  AD.      Then  the  value  oi 

BCFE  =  a  X  ^^1^,  and  the  value  of  ADFE  =  6  X  ?-^— . 

Putting  the  sum  of  these  equal  to  the  value  required  to  be  parted  off,  we  obtain 
__  value  required  X  10 
*  "  ~aXBE  +  bXAE  ' 

(625)  All  the  constructions  of  this  article  depend  on  the  equivalency  of  trian- 
gles which  have  equal  bases,  and  lie  between  parallels.  The  length  of  AD  is  de- 
rived from  the  area  of  a  triangle  being  equal  to  its  base  by  half  its  altitude. 

(527)  Since  similar  triangles  are  to  each  other  as  the  squares  of  their  homolo- 
gous sides, 

ABC  :  DBE  : :  AB" :  BD^ ;  whence  BD  =  AB  a/  5^  =  aB  a/    ^     . 

y    ABC  y  m-\-  n 

The  construction  of  Fig.  363  is  founded  on  the  proportion 

BF  :  BG  : :  BG  :  BA ;  when  BD  =  BG  =  v'  (BA  X  BF)  =  BA  a/    "^     . 

y   m-\-  n 

(52§)  By  hypothesis,  AEF  :  EFBC  ■.■.m:n;  whence  AEF  :  ABC  ::»«:»«  +  »» 

and  AEF  =  ABC  -^  =  ^^2i^  .  _^.      Also,  AEF  =  i  •  AE  X  EF. 
m  -f-  »  "  m  -\-  n 

"rvT>   y    A  "P 

The  similar  triangles  AEF  and  ABD  give  AD  :  DB  : :  AE  :  EF  =  — .    The 

DB  X  AE 

second  expression   for  AEF  then  becomes  AEF  =  ^  AE  • — - — .     Equating 

this  with  the  other  value  of  AEF,  we  have 


AC  X  DB        m  AE*  X  DB 


;  whence  AE  =  i/C  AC  X  AD  X      *?     V 
y    \  'III  -\-  nf 


2  m  +  n  2 .  AD      '  T    V  m -\- 

(530)  In  Fig.  366,  the  triangles  ABD,  DBC,  having  the  same  altitude,  are  to 
each  other  as  their  bases. 

In  the  next  paragraph,  we  have  ABD  :  DBC  : .  AD  :  DC  : :  m  :  n  ;  whence 
AD  :  AC  ::  m  :  7n-\-n;  and  AC  ;  DC  : :  7n -{- n  :  n  ;  whence  the  expressions  for 
AD  and  DC. 

In  Fig.  367,  the  expression  for  AD  is  given  by  the  proportion  AD  :  AC : :  w  :  m  -f  «• 
Similarly  for  DE,  and  EC. 

(531)  In  Fig.  368,  conceive  the  line  EB  to  be  drawn.  The  triangle 
AEB  =  i  ABC,  having  the  same  altitude  and  half  the  base  ;  and  AFD  ^  AEB, 
Decause  of  the  equivalency  of  the  triangles  EFD  and  EFB,  which,  with  AEF,  make 
up  AFD  and  AEB. 

The  point  F  is  fixed  by  the  similar  triangles  ADB  and  AEF 

The  expression  for  AF,  in  the  last  paragraph,  is  given  by  the  proportion, 

ABC  :  ADF  : :  AB  X  AC  :  AD  X  AF  ; 

. .,.,      AB  X  AC    ADF      AB  X  AC        »« 
whence,  AF : 


AD  ABC  AD  m  +  n 

(532)  The  areas  of  triangles  being  equal  to  the  product  of  their  altitudes  by 
half  their  bases,  the  constructions  in  Fig.  369  and  Fig.  370  follow  therefrom. 


406  DEI»IOXSTRATIO\S  [app.  b. 

(533)  In  Fig.  371,  conceive  tlie  line  BL  to  be  drawn.  The  triangle  ABL  will 
be  a  third  of  ABC,  having  the  same  altitude  and  one-third  the  base  ;  and  AED  is 
equivalent  to  ABL,  because  ELB  =-  ELD,  and  AEL  is  common  to  both.  A  similar 
proof  applies  to  DCG. 

(534)  In  Fig.  372,  the  four  smaller  triangles  are  mutually  equivalent,  because 
of  their  equal  bases  and  altitudes,  two  pairs  of  them  lying  between  parallels. 

(535)  In  Fig.  373,  conceive  AE  to  be  drawn.  The  triangle  AEC  =  i.  ABC, 
having  the  same  altitude  and  half  the  base ;  and  EDFC  =  AEC,  because  of  the 
common  part  FEC  and  the  equivalency  of  FED  and  FEA. 

(536)  In  Fig.  374,  in  addition  to  the  lines  used  in  the  problem,  draw  BF  and 
DG.  The  triangle  BFC  ^  i  ABC,  having  the  same  altitude  and  half  the  base. 
Also,  the  triangle  DFG  =  DFB,  because  of  the  parallels  DF  and  BG.  Adding  DFC 
to  each  of  these  triangles,  we  have  DCG  =  BFC  =  i  ABC.  We  have  then  to 
prove  LMC  ^  DCG.  This  is  done  precisely  as  in  the  demonstration  of  Art.  (519), 
page  402. 

(537)  Let  AE  =  a;,  ED  =  y,  AH  =  x',  HF  =  ?/',  AK  =  a.,  KB  =  b. 

The  quadrilateral  AFDE,  equivalent  to  ^  ABC,  but  which  we  wiU  represent; 
generally,  by  m^,  is  made  up  of  the  triangle  AFH  and  the  trapezoid  FHED. 

AFH  =  i .  z'y'.  FHED  =  i  [x  —  x')  {y  +  y"). 

.-.  AFDE  =  m2  =  I .  x'y'  +  i{x  —  x')  {y-\-y')=ix{y  +  y')  —  \  x'y. 
The  similar  triangles,  AHF  and  AKB,  give 

a:  ow  X  :  y  =  — . 

Substituting  this  value  of  y'  in  the  expression  for  m"^,  we  have 

m'^=^x  ly+  —J  —ix'y; 

,  _  a  (2  »m2  _  a;^)  _  AK  (f  ABC  —  AE  X  ED) 
w  ence,  x  j^ZT"^^  —  KB  X  AE  — AKxED' 

The  formula  is  general,  whatever  may  be  the  ratio  of  the  area  m^  to  that  ol 
the  triangle  ABC. 

(53§)  In  Fig.  376,  FD  is  a  line  of  division,  because  BF  =  the  triangle  BDF 
divided  by  half  its  altitude,  which  gives  its  base.    So  for  the  other  triangles. 

(539)  In  Fig.  377,  D6  is  a  second  line  of  division,  because,  drawing  BL,  tha 
triangle  BLC  =  J  ABC ;  and  BDGC  is  equivalent  to  BLC,  because  of  the  common 
part  BCLD,  and  the  equivalency  of  the  triangles  DLG  and  DLB. 

To  prove  that  DF  is  a  third  line  of  division,  join  MD  and  MA.  Then 
BMA  =  i  BGA.  From  BM  A  take  MFA  and  add  its  equivalent  MFD,  and  we  have 
MDFB  =  i  BGA  =  i  (ABDG  —  BDG)  =  i  (§  ABC  —  BDG)  =  §  ABC  —  i  BDQ, 

To  MDFB  add  MDB,  and  add  its  equivalent,  |  BDG,  to  the  other  side  of  the  equa- 
tion, and  we  have 

MDFB  +  MDB  =  i  ABC  — iBDG  +  iBDG;  or,  BDF  =  i  ABC. 

(540)  In  Fig.  378,  the  triangle  AFC  =  J  ABC,  having  the  same  base  and  one- 
third  the  altitude.  The  triangles  AFB  and  EFC  are  equivalent  to  each  othei; 
each  being  composed  of  two  triangles  of  equal  bases  and  altitudes ;  and  each  if 
therefore  one  third  of  ABC. 


Ai'p.  B.]  For  Part  XI.  407' 

In  Fig.  379,  AFC  :  ABC  : :  AD  :  AB ;  since  these  two  triangles  have  the  common 
base  AC,  and  their  altitudes  are  in  the  above  ratio.  So  too,  BFC :  ABC : :  BE  :  BA, 
Hence,  the  remaining  triangle  AFB  :  ABC  : :  DE  :  AB. 

(541)  By  Art.  (65),  Note,  ABC  =  ^  AC  X  CB  X  sin.  ACB.  But  the  angle 
ACB  =  ACD+DCB  =  i  (180°— ADC)+i  (180°-CDB)  =  180°-i  (ADC+CDB). 
Hence,  ABC  =  i  AC  X  CB  X  sin.  j  (ADC  +  CDB)  =  i  AC  X  CB  X  sin.  i  ADR 

Let  r  =  DA  =  DB  =  DC.  Since  AB  is  the  chord  of  ADB  to  the  radius  r,  and 
therefore  equal  to  twice  the  sine  of  half  that  angle,  we  have 

.     ,     .^^       AB  ^Ti     AB         ,         ABXBCXCA 

em.  J .  ADB  =  — -  ;  whence,  ABC  =  A  AC  X  CB  X —  ;  and  r  = 

2  r  ■'  2  r '  4 .  ABC 

Also,  since  the  area  of  each  of  the  three  small  triangles  equals  half  the  product  of 
the  two  equal  sides  into  the  sine  of  the  included  angle  at  D,  these  triangles  will 
be  to  each  other  as  the  sines  of  those  angles.     These  angles  are  found  thus  : 

sin.  i  ADB  =  ^ ;  sin.  *  BDC  =  —  :  sin.  A  ADC  =  — . 
^  2r  ^  2r'  ^  2r 

(542)  The  formulas  in  this  article  are  obtained  by  substituting,  in  those  of  Art 
(523),  for  the  triangle  DBE,  its  equivalent  ■    ^"'      X  i  AB  X  BC  X  sin.  B. 

„„,,       ,  .//     m        ABxBCXsin.  B\  //     m         .„     ^  A 

BD  thus  becomes  =  1/ ( — ;— ^-— I  =4/  I  — ; — XABXBC); 

r     \m  -f-  n  sm.  B  /        r     V/n  -\-n  / 

\^(;;rx7XASxI^CX8in.2B)      gin  g        /.    „i  v 

andDE  =  ^lA!^±^^ >'=  i^  .  ^(-J-XABxBc). 

COS.  i  a  cos.  i  B     r    \w  -f-  ti  / 

(543)  The  rule  and  example  prove  themselves. 

(544)  In  Fig.  383,  conceive  the  sides  AB  and  DC,  produced,  to  meet  in  some 
point  P.     Then,  by  reason  of  the  similar  triangles,  ADP  :  BCP  : :  AD^ ;  BC 
whence,  by  "  division,"  ADP  —  BCP  =  ABCD  :  BCP  : :  AD^  —  BC^ :  BC^. 

In  like  manner,  comparing  EFP  and  BCP,  we  get  EBCF :  BCP  : :  EF^—BC^ :  BC* 
Combining  these  two  proportions,  we  have 

ABCD  :  EBCF  : :  AD^  —  BC» :  EF^  —  BC» ; 
or,  m-^n:m::  AD^  —  BC  :  EF^  —  BC. 

Whence;  (m  +  n)  EF^  —  m .  BC^  —  w  BC^  =  wi .  AD^  —  m  .  BC  • 

'  —  i/{^"'  ^  ^^'  +  ^  X  BCV 
y   \  m-{-  n  /' 

Also,  from  the  similar  triangles  formed  by  drawing  BL  parallel  to  CD,  we  have 

BA  X  EK      AB  (EF  —  BC) 


.-.EF: 


AL : EK ; :  BA : BE  = : 


AL  AD  —  BC 


(545)  Let  BEFC  =  — —  •  ABCD  =  a;     let    BC  =  6  ;     BH  =  A;    and 
m  +  » 
AD  —  BC  =  e.   Also  let  BG  =  a; ;  and  EF  =  y.   Draw  BL  parallel  to  CD.    By  eim 
ilar  triangles,  AL :  EK  : :  BA  :  BE  : :  BH  :  BG ;  or,  AD-BC  :  EF-BC  • :  BH  -.BG ; 

\.  e.,  e  :  y  —  o  ::  h  :  x;  whence  x  =  — — . 

c 

Also,  the  area  BEFC  =  a  =  i .  BG  (EF  +  BC)  =  i  x  (y  +  6) ;  whence  y  = — -4 


i08  DEMOi\STRATIONS.  [a pp.  b 

Substituting  this  value  of  y  in  the  exi^ression  for  x,  and  reducing,  we  obtain 

,   Ibh  lah        ,  ,  bh   ,       //2  ah   ,   b^h^\ 

X  A X  = ;  whence  we  have  a:  = +  a/  I f-  - — -—  I. 

c  c  c         y    \    c  c'   / 

The  second  proportion  above  gives  y  —  b  =—  ;  whence  y  =  6  -f-  —  •  a;. 
Replacing  the  symbols  by  their  lines,  we  get  the  formulas  in  the  text. 

(546)  ABEF=  A  ABCD.  But  ABRP  =  ABEF,  because  of  the  common  part 
ABRF,  and  the  triangles  FRP  and  FEE,  which  make  up  the  two  figures,  and 
which  are  equivalent  because  of  the  parallels  FR  and  PE.     So  for  the  other  parts. 

(547)  The  truth  of  the  foot-note  is  evident,  since  the  first  line  bisects  the  tra- 
pezoid, and  any  other  line  drawn  through  its  middle,  and  meeting  the  parallel 
sides,  adds  one  triangle  to  each  half,  and  takes  away  an  equal  triangle ;  and  thus 
does  not  disturb  the  equivalency. 

(54§)  In  Fig.  885,  since  EF  is  parallel  to  AD,  we  have  ADG  :  EGF : :  GH^ :  GK^. 
EGF   is    made  up  of   the  triangle  BCG  =  a',  and  the    quadrilateral  BEFC  ■=?• 

— ; —  •  ABCD  = j —  •  («  —  «').     Hence  the  above  proportion  becomes 

m-^-  n  m  ■}■  71 

m 

a:a'-\ ; —  (a  —  a')::  Gm  :  GK^ :  or, 

m  +  n 

(m  +  n)a:ma  +  na'  : :  GH» :  GK^ ;  whence  GK  =  GH  a/ (^^^—-\, 

f    \{m-{-n)a/ 

GK 
GE  is  given  by  the  proportion  GH  ;  GK  : :  GA  :  GE  =  GA  •  --— . 

In  Fig.  386,  the  division  into  p  parts  is  founded  on  the  same  principle.     Th«, 

triangle  EFG  =  GBC  +  EFCB  =  a'  -f  — .    Now  ADG  :  EFG  : :  AG^  :  EG^ ; 

P 

or,        a'  +  Q:a'+-::  AG^iEG'^;  whence  GE  =  AG  V   ( —— ^  I . 
p  '      \  a  +  Q  / 

2Q 

GL  is  obtained  by  taking  the  triangle  LMG  =  a'  -|-  —  ;  and  so  for  the  rest. 

(552)  In  Fig.  390,  join  FC  and  GO.  Because  of  the  parallels  CA  and  BF,  the 
triangle  FCD  will  be  equivalent  to  the  quadrilateral  ABCD,  of  which  GCD  will 
therefore  be  one  half;  and  because  of  the  parallels  GE  and  CH,  EHDC  will  be 
equivalent  to  GOD. 

(553)  In  Fig.  391,  by  drawing  certain  lines,  the  quadrilateral  can  be  divided 
into  three  equivalent  parts,  each  composed  of  an  equivalent  trapezoid  and  an 
equivalent  triangle.  These  three  equivalent  parts  can  then  be  transformed,  by 
means  of  the  parallels,  into  the  three  equivalent  quadrilaterals  shown  in  the 
figure.     The  full  development  of  the  proof  is  left  as  an  exercise  for  the  student. 

In  Fig.  392,  draw  CG.  Then  CBG  =  \  ABCD.  But  CKQ  =  CGQ.  Therefore 
CKQB  =  \  AECD.     So  for  the  other  division  line. 

(556)  The  division  of  the  base  of  the  equivalent  triangle,  divides  the  polygon 
similarly.  The  point  Q  results  from  the  equivalency  of  the  triangles  ZBP  and  ZBQ 
PQ  being  parallel  to  BZ, 


APPENDIX    C, 


INTRODUCTION   TO  LEVELLING. 

(1)  The  Principles.  Letzllixg  is  the  art  of  finding  how  much  oue  poiut 
is. higher  or  lower  than  another;  L  e.,  how  much  one  of  the  points  is  above  or  below 
a  level  line  or  surface  which  passes  through  the  other  point. 

A  level  or  horizontal  line  is  one  which  is  perpendicular  to  the  direction  of  grav- 
ity, as  indicated  by  a  plumb-line  or  similar  means.  It  is  therefore  parallel  to  the 
surface  of  standing  water. 

A  level  or  horizontal  surface  is  defined  m  the  same  way.  It  will  be  determined 
by  two  level  lines  which  intersect  each  other.* 

Levelling  may  be  named  Vertical  Surveying,  or  Up-and-down  Surveying ;  the 
subject  of  the  preceding  pages  being  Horizontal  Surveying,  or  Right-and-left  and 
Fore-and-aft  Surveying. 

All  the  methods  of  Horizontal  Surveying  may  be  used  in  Vertical  Surveying. 
The  one  which  will  be  briefly  sketched  here  corresponds  precisely  to  the  method 
of  "  Surveying  by  offsets,"  founded  on  the  Second  Method,  Art.  (6),  "  Rectangular 
Co-ordinates,"  and  fully  explained  in  Arts.  (114),  Ac. 

The  operations  of  levelling  by  this  method  consist,  firstly,  in  obtaining  a  level 
line  or  plane  ;  and,  secondly,  in  measuring  how  far  below  it  or  above  it  (usually 
the  former)  are  the  two  points  whose  relative  heights  are  required. 

(2)  The  Instruments.   A  level  rig.4i5. 

line  may  be  obtained  by  the  following       <J   I ^ j- 

simple  instrument,  called  a  "  Plumb-line  ~ 

level."  Fasten  together  two  pieces  of 
wood  at  right  angles  to  each  other,  so  as 
to  make  a  T,  and  draw  a  line  on  the  up- 
right one  so  as  to  be  exactly  perpendicu- 
lar to  the  top  edge  of  the  other.  Suspend 
a  plumb-line  as  in  the  figure.  Fix  the  T 
against  a  staff  stuck  in  the  ground,  by  a 
ecrew  through  the  middle  of  the  cross-  1 

piece.     Turn  the  T  till  the  plumb-line  ^ 

exactly  covers  the  line  which  was  drawn. 

Then  will  the  upper  edge  of  the  cross-piece  be  a  level  line,  and  the  eye  can  sight 
across  it,  and  note  how  far  above  or  below  any  other  point  this  level  line,  pro- 
.onged,  would  strike.  It  will  be  easier  to  look  across  sights  fixed  on  each  end  of 
the  cross-piece,  making  them  of  horsehair  stretched  across  a  piece  of  wire,  beat 
mto  three  sides  of  a  square,  and  stuck  into  each  end  of  the  cross-piece  ;  taking  care 
Shat  the  hairs  are  at  exactly  equal  heights  abc7e  the  upper  edge  of  the  cross-piece. 

*  Certain  small  corrections,  to  be  hereafter  explained,  will  be  ignored  for  the  present,  and  w« 
iFlll  oonsider  level  lines  as  straight  lines,  and  level  surfaces  as  planes. 


410 


LEVELLIIVG. 


[app.  r 


■i. 


A  modification  of  this  is  to  fasten  a  common  ^'S-  ^^• 

carpenter's  square  in  a  slit  in  the  top  of  a  staff,     ,3^f*-  "'K        fSi" 
by  means  of  a  screw,  and  then  tie  a  plumb-line 
at  the  angle  so  that  it  may  hang  beside  one  arm. 
When  it  has  been  brought  to  do  so,  by  turning 
the  square,  then  the  other  arm  will  be  leveL 

Another  simple  instrument  depends  upon  the 
nrinciple  that  "  water  always  finds  its  level," 
corresponding  to  the  second  part  of  our  defini- 
tion of  a  level  line.  If  a  tube  be  bent  up  at  each 
end,  and  nearly  filled  with  water,  the  surface  of 
the  water  in  one  end  will  always  be  at  the  same 
height  as  that  in  the  other,  however  the  position 

of  the  tube  may  vary.  On  this  truth  depends  the  "  Water-level."  It  may  b« 
easily  constructed  with  a  tube  of  tin,  lead,  copper,  &c.,  by  bending  up,  at  right 
angles,  an  inch  or  two  of  each  end, 

and  -supporting   the    tube,    if    too  ■^'^'      • 

flexible,  on  a  wooden  bar.  In  these  .sp- 
ends cement  (with  putty,  twine 
dipped  in  white-lead,  <fec.),  thin  phi- 
als, with  their  bottoms  broken  off, 
so  as  to  leave  a  free  communication 
between  them.     Fill  the  tube  and 

the  phials,  nearly  to  their  top,  with  colored  water.  Blue  vitriol,  or  cochineal, 
may  be  used  for  coloring  it.  Cork  their  mouths,  and  fit  the  instrument,  by  a 
steady  but  flexible  joint,  to  a  tripod.  Figures  of  joints  are  given  on  page  134,  and 
of  tripods  on  page  133. 

To  use  it,  set  it  in  the  desired  spot,  place  the  tube  by  eye  nearly  level,  remove 
the  corks,  and  the  surfaces  of  the  water  in  the  two  phials  will  come  to  the  same 
leveL  Stand  about  a  yard  behind  the  nearest  phial,  and  let  one  eye,  the  other 
being  closed,  glance  along  the  right-hand  side  of  one  phial  and  the  left-hand  side 
of  the  other.  Raise  or  lower  the  head  till  the  two  surfaces  seem  to  coincide,  and 
this  line  of  sight,  prolonged,  will  give  the  level  line  desired.  Sights  of  equal 
height,  floating  on  the  water,  and  rising  above  the  tops  of  the  phials,  would  give 
a  better-defined  line. 

The  "  Spirit-level"  consists  essentially  Fig.  4iS. 

of  a  curved  glass  tube  nearly  filled  with 
alcohol,  but  with  a  bubble  of  air  left 
within,  which  always  seeks  the  highest 
Bpot  in  the  tube,  and  will  therefore  by 
its  movements  indicate  any  change  in 
the  position  of  the  tube.  Whenever  the  bubble,  by  raising  or  lowering  one  end, 
has  been  brought  to  stand  between  two  marks  on  the  tube,  or,  in  case  of  expan- 
sion or  contraction,  to  extend  an  equal  distance  on  either  side  of  them,  the  bottom 
of  the  block  (if  the  tube  be  in  one),  or  sights  at  each  end  of  the  tube,  previously 
properly  adjusted,  will  be  on  the  same  level  line.  It  may  be  placed  on  a  board 
fixed  to  the  top  of  a  staff  or  tripod. 

When,  instead  of  the  sights,  a  telescope  is  made  parallel  to  the  level,  and  van 
ous  contrivances  to  increase  its  delicacy  and  accuracy,  are  added,  the  instrumenf 
becomes  the  Engineer  s  spirit-leveL 


kvv.  c] 


The  Practice. 


411 


Fis.  419. 


(3)  The  Practice.  By  whichever  of  these  various,  meana  a  level  line 
Das  been  obtained,  the  subsequent  operations  in  making  use  of  it  are  identical 
Since  the  "  water-level"  is  easily  made  and  tolerably  accurate,  we  will  suppoae  it 
to  be  emploj'ed.  Let  A  and  B,  Fig. 
419,  represent  th«  two  points,  the 
difference  of  the  heights  of  which  is 
required.  Set  the  instrument  on 
any  spot  from  which  both  the  points 
can  be  seen,  and  at  such  a  height 
that  the  level  line  will  pass  above 
the  highest  one.  At  A  let  an  assist- 
ant hold  a  rod  graduated  into  feet, 
tenths,  (fee.  Turn  the  instrument  to- 
wards the  staff,  sight  along  the  level 
line,  and  note  what  division  on  the 
staff  it  strikes.  Then  send  the  staff 
to  B,  direct  the  instrument  to  it,  and  note  the  height  observed  at  that  point.  li 
the  level  line,  prolonged  by  the  eye,  passes  2  feet  above  A  and  6  feet  above  R.  the 
difference  of  their  heights  is  4  feet.  The  absolute  height  of  the  level  line  itself  is 
a  matter  of  indifference.  The  rod  may  carry  a  target  or  plate  of  iron,  claspe4  to 
it  so  as  to  slide  up  and  down,  and  be  fixed,  at  will.  This  target  may  be  variously 
painted,  most  simply  with  its  upper  half  red  and  its  lower  half  white.  The  hori- 
zontal line  dividing  the  colors  is  the  line  sighted  to,  the  target  being  moved  up 
or  down  till  the  line  of  sight  strikes  it.  A  hole  in  the  middle  of  the  target  shows 
what  division  on  the  rod  coincides  with  the  horizontal  line,  when  it  has  been 
brought  to  the  right  height. 

If  the  height  of  another  point,  C,  Fig.  420,  not  visible  from  the  first  station,  b* 
required,  set  the  instrument  so  as  to  see  B  and  C,  and  proceed  exactly  as  with  A 


Fig.  420, 


»nd  B.  If  C  be  1  foot  below  B,  as  in  the  figure,  it  will  be  5  feet  below  A  It  it 
were  found  to  be  7  feet  above  B,  it  would  be  3  feet  above  A.  The  comparative 
height  of  a  series  of  any  number  of  points,  can  thus  be  found  in  reference  to  any 
one  of  them. 

The  beginner  in  the  practice  of  levelling  may  advantageously  make  m  his  uot«- 
oook  a  sketch  of  the  heights  noted,  and  of  the  distances,  putting  down  each  as  it 
b  observed,  and  imitating,  as  nearly  as  his  accuracy  of  eye  will  permit,  their  pio- 


112 


LEVELLIiXG. 


[apiv 


portional  dimensions.*  But  .vhen  the  observations  are  numerous,  they  should  be 
kept  in  a  tabular  form  such  as  that  which  is  given  below.  The  names  of  the 
points,  or  "  Stations,"  whose  heights  are  demanded,  are  placed  in  the  first  column ; 
and  their  heights,  as  finally  ascertained,  in  reference  to  the  first  point,  in  the  last 
column.  The  heights  above  the  starting  point  are  marked  +,  and  those  below  it 
are  marked  — .  The  back-sight  to  any  station  is  placed  on  the  line  below  the 
point  to  which  it  refers.  "When  a  back-sight  exceeds  a  fore-sight,  their  difference 
is  placed  in  the  column  of  "  Eise ;"  when  it  is  less,  their  difference  is  a  "  Fall." 
The  following  table  represents  the  same  observations  as  the  last  figure,  and  their 
«areful  comparison  will  explain  any  obscurities  in  either. 


Stations. 

Distances. 

Back-sights. 

Fore-sights. 

Eise. 

Fall. 

Total  Heightfl. 

A 

0.00 

B 

100 

2.00 

6.00 

-  4.00 

-4.00 

C 

60 

3.00 

4.00 

-  1.00 

-  5.00 

D 

40 

2.00 

1.00 

+  1.V. 

-  4.00 

E 

10 

6.00 

1.00 

+  5.00 

+  1.00 

F 

50 

2.00 

6.00 

-  4.00 

-  8.00 

15.00 

18.00 

-  3.00 

Tile  above  table  shows  that  B  is  4  feet  below  A  ;  that  C  is  5  feet  below  A ;  that 
E  is  1  foot  above  A ;  and  so  on.  To  test  the  calculations,  add  up  the  back-sights 
and  fore-sights.     The  difference  of  the  sums  should  equal  the  last  "  total  laeight." 

Another  form  of  the  levelling  field-book  is  presented  below.  It  refers  to  the 
same  stations  and  levels,  noted  in  the  previous  form,  and  shown  in  Fig.  420. 


Stations. 

Distances. 

Back-sights. 

Ht  Inst,  above  Datum. 

Fore-sights. 

Total  Heights. 

A 

0.00 

B 

100 

2.00 

+  2.00 

6.00 

-  4.00 

C 

60 

8.00 

-  1.00 

4.00 

—  5.00 

D 

40 

2.00 

-  3.00 

1.00 

-4.00 

E 

70 

6.00 

+  2.00 

1.00 

+  1.00 

F 

50 

2.00 

+  3.00 

6.00 

-  8.00 

15.00 

18.00 

-  3.00 

In  the  above  form  it  will  be  seen  that  a  new  column  is  introduced,  containing 
the  Height  of  the  Instrument  (i.  e.,  of  its  line  of  sight),  not  above  the  ground 
where  it  stands,  but  above  the  Datum,  or  starting-point,  of  the  levels.  The  former 
columns  of  "  Rise"  and  "  Fall"  are  omitted.  The  above  notes  are  taken  thus : 
The  height  of  the  starting-point  or  "  Datum,"  at  A,  is  0.00.  The  instrument  being 
set  up  and  levelled,  the  rod  is  held  at  A.  The  back-sight  upon  it  is  2.00 ;  there- 
fore the  height  of  the  instrument  is  also  2.00.  The  rod  is  next  held  at  B.  The 
fore-sight  to  it  is  6.00.  That  point  is  therefore  6.00  below  the  instrument,  or 
2.00  —  6.00  :=  —  4.00  below  the  datum.  The  instrument  is  now  moved,  and  again 
•et  up,  and  the  back-sight  to  B,  being  3.00,  the  Ht.  Inst,  is  —  4.00  +  8.00  =  — 1.00 


•  In  the  figure,  the  limits  of  the  page  have  made  it  necessary  to  contract  the  horizontal  distanofw 
to  one-tenth  of  their  proper  prov-ortional  size. 


ipp.  c.|  The  Practice.  413 

»nd  80  on :  the  Ht.  Inst,  being  always  obtained  by  adding  the  back-sight  to  tlio 
height  of  the  peg  on  which  the  rod  is  held,  and  the  height  of  the  next  peg  being 
obtained  by  subtracting  the  fore-sight  to  the  rod  held  on  that  peg,  from  the  Ht.  Inst, 

The  level  lines  given  by  these  instruments  are  all  lines  of  apparent  level,  and 
not  of  true  level,  which  should  curve  with  the  surface  of  the  earth.  These  level 
lines  strike  too  high;  but  the  difference  is  very  small  in  sights  of  ordinary  length, 
being  only  one-eighth  of  an  inch  for  a  sight  of  one-eighth  of  a  mile,  and  diminishing 
as  the  square  of  the  distance ;  and  it  may  be  completely  compensated  by  setting 
the  instrument  midway  between  tire  points  whose  difference  of  level  is  desired;  a 
precaution  which  should  always  be  taken,  when  possible. 

It  may  be  required  to  show  on  paper  the  ups  and  downs  of  the  line  which  has 
been  levelled ;  and  to  represent,  to  any  desired  scale,  the  heights  and  distances  of 
the  various  points  of  a  line,  its  ascents  and  descents,  as  seen  in  a  side-view.  This 
is  called  a  "  Profile."  It  is  made  thus.  Any  point  on  the  paper  being  assumed 
for  the  first  station,  a  horizontal  line  is  drawn  thi-ough  it ;  the  distance  to  the  next 
station  is  measured  along  it,  to  the  required  scale  ;  at  the  termination  of  this  dis- 
tance a  vertical  line  is  drawn ;  and  the  given  height  of  the  second  station  above  or 
below  the  first  is  set  off  on  this  vertical  line.  The  point  thus  fixed  determines 
the  second  station,  and  a  line  joining  it  to  the  first  station  represents  the  slope  of 
the  ground  between  the  two.     The  process  is  repeated  for  the  next  station,  &c. 

But  the  rises  and  falls  of  a  line  are  always  very  small  in  proportion  to  the  dis 
tances  passed  over  ;  even  mountains  being  merely  as  the  roughnesses  of  the  rind 
of  an  orange.  If  the  distances  and  the  heights  were  represented  on  a  profile  to  the 
same  scale,  the  latter  would  be  hardly  visible.  To  make  them  more  apparent  it 
is  usual  to  "  exaggerate  the  vertical  scale"  ten-fold,  or  more ;  i.  e.,  to  make  the 
representation  of  a  foot  of  height  ten  times  as  great  as  that  of  a  foot  of  length,  as 
in  Fig.  420,  in  which  one  inch  represents  one  hundred  feet  for  the  distances,  and 
ten  feet  for  the  heights. 

Tlie  preceding  Introduction  to  Levelling  has  been  made  as  brief  as  possible ,  but 
by  any  of  the  simple  instruments  described  in  it,  and  either  of  its  tabular  forms,  any 
person  can  determine  with  sufiicient  precision  whether  a  distant  spring  is  higher  or 
lower  than  his  house,  and  how  much  ;  as  well  as  how  deep  it  would  be  necessary 
to  cut  into  any  intervening  hill  to  bring  the  water.  He  may  iii  like  manner  ascer- 
tain whether  a  swamp  can  be  drained  into  a  neighboring  brook  ;  and  can  cut  the 
necessary  ditches  at  any  given  slope  of  so  many  inches  to  the  rod,  (fee.,  having  thus 
found  a  level  line ;  or  he  can  obtain  any  other  desired  information  which  depend.* 
on  the  relative  heights  of  two  points. 

To  explain  the  peculiarities  of  the  more  elaborate  levelling  instruments,  the 
precautions  necessary  in  their  use,  the  prevention  and  correction  of  errors,  the 
overcoming  of  difficulties,  and  the  various  complicated  details  of  their  applications, 
would  require  a  great  number  of  pages.  This  will  therefore  be  reserved  for  an- 
other volume,  as  announced  in  the  Preface. 


414 


APPEKDIX  D. 


MAGNETIC    VARIATIONS 

IN  THE  UNITED  STATES. 

[From  a  Report  by  C.  A.  SCHOTT,  Assistant  U.  S   Coast  Survey].    See  Silliman's 
Journal,  May,  1860,  p.  335 ;  and  U.  S.  Coast  Survey  Report  for  185y,  App.  24,  p.  296. 


W.  and  E.  indicate  West  and  East  Declinations.   They  are  giv67i  below  in  DegreeH  and  tenths. 


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i  n  e  d  . 

AMLYTICAL  TABLE  OF  CONTENTS. 


PART     I. 

GENERAL  PROCIPLES  AND  FUNDAMEXTAL  METHODS. 

CHAPTER  I.    Definitions  and  Methods. 


^1)  Surveying  defined 9 

(2)  When  a  point  is  determined. .  9 

(3)  Determining  lines  and  surfaces  10 

To  determine  jtoints. 

(5)  First  Method 10 

(6)  Second       do 11 

(7)  Third        do 11 

(§)  Fourth      do.'. 12 

(10)  Fifth      do 13 


ABTICUB  P^oa 

Division  of  the  subject. 

(12)  By  the  methods  employed  . .  14 

(13)  By  the  instruments 14 

(14)  By  the  objects U 

(15)  By  the  extent 15 

(16)  Arrangement  of  this  book. . .  15 

(17)  The  three  operations  common 

to  all  surveying 16 


CHAPTER  II.    Wakinff  the  Measurements. 


Measuring  straight  lines. 

(19)  Actual  and  Visual  lines  ....  16 

(20)  Gunter's  Chain 16 

(21)  Pins 19 

(22)  Staves 19 

(23)  How  to  chain 19 

(24)  Tallies 21 


(25)  Chaining  on  slopes 21 

(2§)  Tape 23 

(29)  Rope,  &C. 24 

(30)  Rods 24 

(32)  Measuring-wheel 24 

(33)  Measuring  Angles    25 

(34)  Noting  the  Measurements  ...  25 


CHAPTER  III.    Drawing  the  Map. 


(35)  A  Map  defined 25 

(36)  Platting 25 

(37)  Straight  lines 26 

(3§)  Arcs 26 

(39)  Parallels 26 

(40)  Perpendiculars 27 

(41)  Angles 28 

(42)  Drawing  to  scale 28 

(44)  Scales 29 


(45)  Scales  for  farm  surveys 29 

(46)  Scales  for  state  surveys 31 

(47)  Scales  for  railroad  surveys  . .  32 

(49)  How  to  make  scales 33 

(50)  The  Vernier  scale 35 

(51)  A  reduced  scale 36 

(52)  Sectoral  scales 36 

(53)  Drawing  scale  on  map 37 

(54)  Scale  omitted 37 


CHAPTER  IV.    Calculatinff  the  Content. 


(55)  Content  defined 

(56)  Horizontal  measurement. . . . 

(57)  Unit  of  content 

(58)  Reductions 

(59)  Table  of  Decimals  of  an  acre. 

(60)  Chain  correction 

(61)  Boundary  lines 


Mfthods  of  Calculation. 

(63)  First  Method,  Arithmetically. 

(64)  Rectangles 

(65)  Triangles 

(66)  Parallelograms 

(67)  Trapezoids 


38 

(6§) 

38 

(«9) 

40 

(TO) 

40 

(71) 

41 

(72) 

41 

{T3) 

42 

('S'4) 

(75) 

(7«) 

43 

(77) 

43 

(7S) 

43 

(S4) 

44 

(»7) 

44 

Quadrilaterals 44 

Curved  boundaries 45 

Second  Method,  Geometrically  45 

Division  into  triangles 45 

Graphical  multiplication ...  47 

Division  into  trapezoids. ...  48 

Do.      into  squares 48 

Do.      into  parallelograms  49 

Addition  of  widths 5i) 

Tliird  Method,  InstmmeniaUy  50 

Reduction  to  one  triangle . .  50 

Special  instruments 54 

Fourth  Method,  2rigono7netri- 

cally :  68 


416 


CONTENTS. 


PART    II. 


CHAIN  SURVEYING. 
CHAPTER  I.    Surveying  by  Diagonals. 


ASTIOLB  PASS 

(90)  A  three-sided  field 58 

(91)  A  four-sided  field 59 

(92)  A  many-sided  field 60 

(93)  How  to  divide  a  field 61 


Keeping  the  field-notes 62 

(94)  By  sketch 62 

(95)  In  columns 62 

(96-9T)  Field-books 64 


CHAPTER  II.    Surveying  l)y  Tie-lines. 


(9§)  Surveying  by  tie-lines  ....     66 
(100)      Chain  angles 67 


(101) 
(102) 


Inaccessible  areas 67 

Without  platting 67 


CHAPTER  III.    Surveying  by  Perpendiculars. 


To  set  out  Perpendiculars. 

(104)    By  Surveyor's  Cross 69 

(lOT)     By  Optical  Square 70 

(108)     By  the  Chain 72 

Diagonals  and  Perpendiculars, 

(110)  A  three-sided  field 72 

(111)  A  four-sided  field 73 

(112)  A  many-sided  field 74 

(113)  By  one  diagonal 75 


(114) 
(117) 
(118) 
(11») 
(120) 

(121) 
(122) 
(123) 
(124) 


Taking  offsets 75 

Double  offsets 76 

Field  work 77 

Platting 79 

Calculating  content 80 

When  equidistant 80 

Erroneous  rules 81 

Reducing  to  one  triangle  81 

Equalizing 81 


CHAPTER  IV.    Surveying  by  the  methods  combined. 


(125)  Combination   of  the   three 

preceding  methods 82 

(127)  Field-books 83 

(130)  Calculations 88 

(131)  The  six-line  system 90 


(132)  Exceptional  cases 92 

(134)  Inaccessible  areas 93 

(136)  Roads 95 

(137)  Towns 95 


CHAPTER  V.    Obstacles  to  Measurement  in  Chain  Surveying. 

(138)  The  obstacles  to  Ahnement  and  Measurement 96 

(139)  Land  Geometry 96 

Problems  on  Perpendiculars. 

(140)  Problem  1.  To  erect  a  perpendicular  at  any  point  of  a  line 97 


(143) 

(148) 
(150) 
(153) 

(156) 
(158) 


when  the  point  is  at  or  near  the 

end  of  the  line 98 

"  "  when  the  line  is  inaccessible  ...     99 

To  let  fall  a  perpendicular  from  a  given  point  to  a  given  line 
"  "  when  the  point  is  nearly  oppo- 

site to  the  end  of  the  line. . . 
"  "  when  the  point  is  inaccessible. . 

"  "  when  the  line  is  inaccessible . . . 


99 


100 
101 
101 


Problems  on  Parallels. 


(160)  Problem  1.  To  run  a  line  from  a  given  point  parallel  to  a  given  line.   102 
(165)  2.  Do  when  the  line  is  inaccessible 103 


CONTEXTS. 


417 


(169) 

(171) 
(172) 
(173) 


(177) 
(17§) 
(179) 
(180) 


(186) 
(1§7) 
(188) 


(191) 
(194) 
(195) 
(196) 


Obstacles  to  Alinement. 
A.  To  ■prolong  a  line 105 


By  ranging  -with  rods 105 

By  perpendiculars 106 

By  equilateral  triangles  . . .    106 
By  symmetrical  triangles. .   107 


(174)  By  transversals 107 

(175)  By  barmonic  conjugates. . .   lo8 

(176)  By  the   complete    quadri- 

lateral   108 


B.  To  interpolate  points  in  a  line 109 


Signals 109 

Ranging 109 

Across  a  valley 110 

Over  a  hill 110 


(1§1)  With  a  single  person Ill 

(1§2)  On  water Ill 

(183)  Through  a  wood 112 

(184)  To  an  invisible  intersection.  112 


Obstacles  to  Measurement. 
A.    \Mien  both  ends  of  the  line  are  accessible 113 


(189)  By  transversals 114 

(190)  In  a  town 114 


By  perpendiculars 113 

By  equilateral  triangles. . .   113 
By  symmetrical  triangles. .   114 

B.    ^Vhen  one  end  of  the  line  is  inaccessible ! 115 


By  perpendiculars 115 

By  parallels 116 

By  a  parallelogram 116 

By  symmetrical  triangles. .  116 


(198)  By  transversals 117 

(199)  By  harmonic  division 117 

(200)  To  an  inaccessible  line 118 

(201)  To  an  inacc.  intersection  ..  118 


(202) 
(203) 


(207) 

C208) 


C.   When  both  ends  of  the  line  are  inaccessible 119 

By  similar  triangles 119  1  (204)  By  a  parallelogram 119 

By  parallels 119  |  (205)  By  symmetrical  triangles. .   120 

INACCESSIBLE  AREAS 1-21 

Triangles 121  I  (209)  Polygons 121 

Quadrilaterals 121  [ 


PART    III. 


COMPASS   SURVEYING. 
CHAPTER  I.    Angular  SnrTeying  in  general. 


(210)  Principle 122 

(211)  Definitions 122 

(213)  Goniometer 123 

(214)  How  to  use  it 123 

'215)  Improvements 124 


(217)  The  Compass 124 

(219)  Methods   of   Angular  Sur- 

veying     126 

(220)  Subdivisions  of  Polar  Sur- 

veying    125 


CHAPTER  II.    The  Compass. 


(221) 
(222) 
(223) 
(224) 
(225) 
(226) 
(227) 


The  Needle 127 

The  Sights 128 

The  Telescope 128 

The  divided  Cu-cle 128 

The  Points 129 

Eccentricity 180 

Levels 132 


(228)  Tangent  Scale 132 

(229)  The  Vernier 132 

(230)  Tripods ISS 

(231)  Jacob's  Staff 1S4 

(232)  The  Prismatic  Compass...   135 
(234)  The  defects  of  the  Comfass,  187 


21 


4-18 


fOMEXTS. 


CHAPTER  III.    The  Field-work. 


AKTlrC.^  PAGE 

(235)  Taking  Bearings 138 

(236)  Why  E.  and  W.  are  re- 

versed  '.    139 

(237)  Reading  with  Vernier. .    140 
(23§)         Practical  Hints 140 

Mark  stations.  Set  beside 
fence.  Level  crossways.  Do 
not  level  by  needle.  Keep 
same  end  ahead.  Read  from 
same  end.  Caution  in  read- 
ing. Cheek  vibrations.  Tap 
compass.  Keep  iron  away. 
Electricity.  To  carry  com- 
pass.   Kxtra  pin  and  needle. 


(239) 
(240) 
(241) 


To  magnetize  a  Needle.  142 

Back-sights 143 

Local  Attraction 143 


ABTIOLK  rA«i 

(242)  Anglee  of  deflection .        .    144 

(243)  Angles  between  courses    . .   146 

(244)  To  change  Bearings    146 

(245)  Line  Surveying  14t 

(246)  Checks  by  intersecting 

bearings 148 

(247)  Keeping  the  Field-notes  149 

(251)  New  York  Canal  Maps.   149 

(252)  Farm  Surveying 150 

(254)       .Field-notes 151 

(256)         Tests  of  accuracy 153 

(258)  Method  of  Radiation .. .    154 

(259)  Method  of  Intersection  .    154 

(260)  Running  out  old  lines. .   154 

(261)  Town  Surveying 155 

(262)  Obstacles  iu  Compass  Sur- 

veying   156 


CHAPTER  IV.    Platting  the  Survey. 


(263)  Platting  in  general 

(264)  With  a  protractor 

(265)  Platting  bearings 

(26§)  To  make  the  plat  close. . 

(269)  Field  platting 

(272)  With  a  paper  protractor 


157 

157 
158 
161 
162 
164 


(273)  Drawing-board  protractor  .   166 

(274)  With  a  scale  of  chords 166 

(275)  With  a  table  of  chords 167 

(276)  With  a  table  of  natural  sines  .168 

(277)  By  Latitudes  and  Depart- 

ures    163 


CHAPTER  V.    Latitudes  and  Departures. 


(27§)  Definitions 169 

(279)  Calculation     of    Latitudes 

and  Departures 170 

(280)  Formulas 171 

(281)  Traverse  Table 171 


Applications. 

(282)  Testing  survey 175 

(283)  Supplying  omissions 176 

(284)  Balancing 177 

(285)  Platting 178 


CHAPTER  ¥1.    Calculating  the  Content. 


(286)  Methods 180 

(287)  Definitions 180 

(288)  Longitudes 181 

(289)  Areas 182 

(290)  A  three-sided  field 182 

(291)  A  four-sided  field 183 


292)  General  rule 184 

293)  To  find  east  or  west  station  184 
(294)  Example  1 184 

(296)  Examples  2  to  13 186 

(297)  Mascheroni's'Theorem 188 


CHAPTER  VII.    The  Variation  of  the  Magnetic  Needle. 


(298)  Definitions 189 

(299)  Direction  of  the  needle 189 

To  determine  the  true  meridian. 

(300)  By  equal  shadows  of    the 

sun 190 

(301)  By  the  North  Star  when  in 

the  meridian 191 

(302)  Times  of  crossing  the  me- 

ridian     193 

(303)  By  the  North  Star  when  at 

its  extreme  elongation. .   194 

(304)  Table  of  times 195 

(305)  Observations 196 


(306)  Table  of  Azimuths 196 

(307)  Setting  out  the  meridian. .  197 
To  determine  the  variation. 

(308)  By  the  bearing  of  the  star.  198 

(309)  Other  methods 199 

(310)  Magnetic  variation   in   the 

United  States 199 

Line  of  no  variation 199 

Lines  of  equal  variation . .  200 

Magnetic  Pole 200 

(31 1)  To  correct  magn.  bearings.  200 

(312)  To  survey  a  line  with  true 

bearings 209 


COiXTENTS. 
CHAPTER  VIII.    Changes  in  the  Variation. 


419 


4KTICLK  PAOB 

(314)  Irregular  changes 2u3 

(315)  The  Diurnal  change 203 

(316)  The  Annual  Change 204 

(317)  The  Secular  change 204 

Tables  for  United  States.  205 
To  determine  the  secular 
change 205 


ARTICLE  PAaa 

(31  §)         By  interpolation 206 

(319)  By  old  hnes 206 

(320)  Effects  of  this  change 207 

(321)  To  run  out  old  lines 208 

(322)  Example 208 

(323)  Remedy  for  the  evils   of 

the  secular  change 210 


PART     IV. 
TRANSIT  AXD  THEODOLITE  SURVEYING. 

(BY  THE  3d  METHOD.) 

CHAPTER  I.    The  Instruments. 


(324) 


(325) 
(326) 
(327) 
(32§) 
(329) 
(330) 
(S31) 
(332) 


General  description  of  the 

Transit  and  Theodohte  . .  211 

The  Transit 212 

The  Theodolite 213 

Distinction  between  them  .  214 

Sources  of  their  accuracy. .  214 

Explanation  of  the  figures .  216 

Sectional  view 216 

Telescopes 217 

Cross  hairs 218 

Instrumental  parallax 220 

Eye-glass  and  object-glass..  221 


(333)  Supports 221 

(334)  Tlie  Indexes.     Eccentricity.  221 

(335)  The  graduated  circle 223 

(336)  Movements.       Clamp     and 

Tangent  screw 223 

(337)  Levels 224 

(33§)  Parallel  plates 225 

(339)  Watch  Telescope 226 

(340)  The  Compass 226 

(341)  Theodolites 226 

(342)  Goniasmometre 227 


(343)  Definition 228 

(344)  Illustration 228 

(345)  General  rules 229 

(346)  Reti'ograde  Verniers 230 

(347)  Illustration 231 

(34§)  Mountain  Barometer 231 

(349)  Circle  divided  into  degrees.   232 

(350)  Circle  divided  to  30' 233 


CHAPTER  II.    Verniers. 

(351)  Circle  divided  to  20' 235 

(352)  Circle  divided  to  15' 236 

(353)  Circle  divided  to  10' 237 

(354)  Reading  backwards 237 

(355)  Arc  of  excess 238 

(356)  Double  Verniers 238 

(357)  Compass  Verniers 239 


CHAPTER  III.    Adjustments. 


(35§)  Their  object  and  necessity  .   240 

(359)  The  three  requirements  in 

the  Transit 240 

(360)  First  Adjuitment.  To  cause 

the  circle  to  be  horizontal 

in  every  position 241 

Verification 241 

Rectification 241 

(361)  Second     Adjustment.       To 

cause  the  line  of  collima- 
tion  to  revolve  in  a  plane    242 
Verification 242 


Rectification 243 

(362)  In  the  Theodolite 245 

(363)  Third  Adjustment.  To  cause 

the  line  of  collimation  to 
revolve  in  a  vertical  plane  246 

Verification  (plumb-line; 
star ;  steeple  and  stake)  246 

Rectification 246 

(364)  Centring  eye-piece 247 

(365)  Centring  object-glass _.  247 

Adjusting  line  of  colli- 
mation   241 


i20 


€OATEi\TS. 


CHAPTER  IV. 

ABTIOLB  PAOB 

(366)  To    measure     a    horizontal 

angle 250 

(367)  Reduction  of  high   and 

low  objects 251 

(368)  Notation  of  angles 252 

(369)  Probable  error 252 

(370)  To  repeat  an  angle 252 

(371)  Angles  of  deflection 253 


The  Field-work. 

ARTIOLE  PAOa 

(372)  Line-surveying 254 

(373)  Traversing,  or  surveying 

by  the  back  angle. . .  254 

(374)  Use  of  the  Compass. . . .  255 

(375)  Measuring  distances  with 

a  telescope  and  rod. .  256 

(376)  Ranging  out  Unes 257 

(377)  Farm-surveying 258 

(37§)  Platting ,  259 


PART   y. 


TRIANGULAR    SURVEYING. 

(BY  THE  4th  METHOD.) 


(379) 
(3§0) 
(3§1) 


(3§3) 
(3§3) 


(384) 


Principle 260 

Outline  of  operations 260 

Measuring  a  base 261 

Materials 261 

Supports 262 

Alinement 262 

Levelling 262 

Contacts 262 

Corrections  of  Base 263 

Choice  of  stations 263 

U.   S.    Coast  Survey  Ex- 
ample    265 

Signals 266 


(385)  Observations  of  the  angles. ,   257 

(386)  Reduction  to  the  centre  . . .   268 

(387)  Correction  of  the  angles  . . .   270 

(388)  Calculation  and  platting. . .   270 

(389)  Base  of  Verification 271 

(390)  Interior  filling  up 271 

(391)  Radiating  Triangulation  .. .   272 

(392)  Farm  Triangulation 272 

(393)  Inaccessible  Areas 273 

(394)  Inversion    of     the    Fourth 

method 273 

(395)  Defects  of  the  Method  of  In- 

tersections     274 


PART    VI. 
TRILINEAR  SURVEYING. 

(BY  THE  5th  METHOD.) 
(396)  The  Problem  of  the  three 


points 275 

^397)  Geometrical  Solution 275 


(398)  Instrumental  Solution 277 

(399)  Analytical  "         277 

(400)  Maritime  Surveying 278 


(402) 
(403) 
(404) 
(405) 
(406) 
(407) 
(408) 


PART     VII. 

OBSTACLES  IN  ANGULAR  SURVEYING. 

CHAPTER  I.    Perpendiculars  and  Parallels. 

To  erect  a  perpendicular  to  a  line  at  a  given  point 279 

To  erect  a  perpendicular  to  an  inaccessible  line,  at  a  given  point  of  it  280 

To  let  fall  a  perpendicular  to  a  line,  from  a  given  point 280 

To  let  fall  a  perpendicular  to  a  line,  from  an  inaccessible  point 280 

To  let  fall  a  perpeadicular  to  an  inaccessible  line  from  a  given  point..  281 

To  trace  a  line  through  a  given  point  parallel  to  a  given  line 281 

To  trace  a  line  through  a  given  point  parallel  to  an  inaccessible  line. .  281 


CONTENTS. 


421 


(409) 
(410) 

(411) 
(412) 


(415) 
(416) 
(417) 


CHAPTER  II.    Obstacles  to  Alincraent. 

A.  To  prolong  a  line 28"i 

General  method 282  (413)  When  the  line  to  be  pro- 

By  perpendiculars 282  longed  is  inaccessible  ...   283 

By  an  equilateral  triangle  .  282  (414)  To  prolong  a  line  with  only 

By  triangulation 283  an  angular  instrument. . .    283 

B.  To  interpolate  points  in  a  line 284 

General  method 284  I  (41§)  By  Latitudes  and  Depart- 
By  a  random  line 284  |  vires,  with  transit 285 

By  Latitudes  and  Depart-  I  (419)  By  similar  triangles 286 

ures,  with  compass 285  |  (420)  By  triangulation 286 


CHAPTER  III.    Obstacles  to  Measurement. 

A.  WJien  both  ends  of  the  liiie  are  accessible 287 

(421)  Previous  means  , 287  1  (423)  A  broken  base 287 

(422)  By  triangulation 287  |  (424)  By  angles  to  known  points.  288 

B.  When  one  end  of  the  line  is  inaccessible 288 


(425)  By  perpendiculars 288 

(426)  By  equal  angles 288 

(427)  By  triangulation 289 

(428)  When  one  point  cannot  be 

seen  from  the  other  ....   289 


(429)  To  find  the  distance  from  a 
1        given  point  to  an  inacces- 
sible line 289 


C.    Wlien  both  ends  of  the  line  are  iiiaccessible 290 


(430)  General  method 290 

(431)  To  measure  an  inaccessible 

distance,  when  a  point  in 

its  line  can  be  obtained . .   291 

(432)  When  only  one  point  can  be 

found  from  which  both 
ends  of  the  line  can  be 
seen 291 


(433)  When  no  point  can  be  found 

from  which  both  ends  can 

be  seen *92 

(434)  To  interpolate  a  base ?92 

(435)  From  angles  to  two  points..  293 

(436)  From  angles  to  three  points  293 

(437)  From  angles  to  four  points.  294 
(43§)  Problem  of  the  eight  points  296 


CHAPTER  IV.    To  Supply  Omissions. 

(439)  General  statement 297 

(440)  Case  1.    When  the  length  and  bearing  of  any  one  side  are  loanting . . . .  298 
Case  2.    When  the  length  of  one  side  and  the  bearing  of  another  are 

wanting 298 

(441)  When  the  deficient  sides  adjoin  each  other 298 

(442)  When  the  deficient  sides  are  separated  from  each  other 299 

(443)  Otherwise  :  by  changing  the  meridian 299 

Case  8.    When  the  lengths  of  two  sides  are  wanting 300 

(444)  When  the  deficient  sides  adjoin  each  other SCO 

(445)  When  the  deficient  sides  are  separated  from  each  other  ....  801 

(446)  Otherwise  :  by  changing  the  meridian 301 

Cask  4    When  the  bearings  of  two  sides  are  wanting 802 

447)  When  the  deficient  sides  adjoin  each  other 802 

(448)  When  the  deficient  sides  are  separated  from  each  other 802 


422 


CONTENTS. 


PART    VIII. 
PLANE   TABLE    SURVEYING. 


Mxncix 

(449)  General  description 303 

(450)  The  Table 303 

(451)  The  Alidade 304 

(452)  Method  of  Radiation 305 

(453)  Method  of  Progression 306 

i.454)  Method  of  Intersection 307 


(455)  Method  of  Resection 308 

(456)  To  Orient  the  Table 308 

(457)  To  find  one's  place  on  the 

ground 309 

(45§)  Inaccessible  distances 310 


PART    IX 


SURVEinC}   WITHOUT  INSTRUMENTS. 


(459)  General  principles 311 

(460)  Distances  by  pacing 311 

(461)  Distances  by  visual  angles.  312 

(462)  Distances  by  visibility 313 


(463)  Distances  by  sound 313 

(464)  Angles 814 

(465)  Methods  of  operation 314 


PART    X. 


MAPPING. 
CHAPTER  I.    Copying  Plats. 


(466)  Necessity 316 

(467)  Stretching  the  paper 316 

(46§)  Copying  by  tracing 317 

(469)  "        on  tracing-paper. .  317 

(470)  "        by  transfer-paper.  317 

(471)  "        by  punctures  ... .  318 

(472)  "        by  intersections  . .  318 

(473)  "        by  squares 319 


(474)  Reducing  by  squares 819 

(475)  "  by     proportional 

scales 320 

(476)  "  by  a  pantagraph  321 

(477)  "  by  a  camera  luci- 

da 321 

(47§)  Enlarging  plats 321 


CHAPTER  II.    Conventional  Signs. 


(479)  Object 322 

(480)  The  relief  of  ground 322 

(4§1)  Signs  for  natural  surface. . .  324 

(4§2)  Signs  for  vegetation 324 


(4§3)  Signs  for  water 325 

(4§4)  Colored  topography 325 

(4§5)  Signs  for  detached  objects.   327 


CHAPTER  III.    Finisliing  tlie  Map. 


(4§6)  Orientation 328 

(487)  Lettermg 328 

(488)  Borders 828 


(489)  Joining  paper 329 

(490)  Mounting  maps 829 


COi\TEMS.  428 

PART    XI. 

LAYING  OCT,  PARTIXG  OFF,  A\D  DIVIDLVG   CP  LA\D. 

CHAPTER  I.    Laying  out  Land. 


ABTIOLK  PAOE 

(491)  Its  object 380 

f492)  To  lay  out  squares 330 

(493)  To  lay  out  rectangles 331 

(494)  To  lay  out  triangles 382 


ARTICLE  PA«> 

(496)  To  lay  out  circles 332 

(497)  Totv-n  lots 333 

(49§)  Land  sold  for  taxes 333 

(499)  New  countries 334 


CHAPTER  IL    Parting  off  Land. 

(500)  Its  object 334 

A.  By  a  line  parallel  to  a  side. 

(501)  To  part  off  a  rectangle 335 

(502)  "         "     a  parallelogram 335 

(503)  "         "     a  trapezoid 335 

B.  By  a  line  perpendicular  to  a  side. 

(505)  To  part  off  a  triangle ' 336 

(507)         "         "      a  quadrilateral 337 

(50§)         "         «      any  figure 337 

C.  By  a  line  running  in  any  given  direction. 

(509)  To  part  off  a  triangle 337 

(511)      "         "   a  quadrilateral 338 

(513)  "         "    any  figure ■ 339 

D.  By  a  line  starting  from  a  given  point  in  a  side. 

(514)  To  part  off  a  triangle 339 

(516)  "         "      a  quadrilateral 340 

(517)  "         "      any  figure 340 

E.  By  a  line  passing  through  a  given  point  within  the  field. 

(519)  To  part  off  a  triangle 342 

(520)  "         "     a  quadrilateral  343 

(522)  "         "     any  figure 344 

F.  By  the  shortest  possible  line. 

(523)  To  part  off  a  triangle 345 

(524)  G.  Band  of  variable  value. 345 

(525)  H.  Straightening  crooked  fences ; .  346 

CHAPTER  lEL    Dividing  up  Land. 

(526)  Arrangement   347 

Division  of  Triangles. 

(527)  By  lines  parallel  to  a  side 347 

'52§)  By  lines  perpendicular  to  a  side 343 

(529)  By  lines  running  in  any  given  direction 348 

(530)  By  lines  starting  from  an  angle 349 

(531)  By  lines  starting  from  a  point  in  a  side 349 

1535)  By  lines  passing  through  a  point  within  the  triangle 861 

540)       Do.      the  point  being  to  be  found 353 

(541)  Do.      the  point  to  be  equidistant  from  the  angles 853 

(542)  By  the  shortest  possible  line 354 

Division  of  Rectangles. 

(543)  By  lines  parallel  to  a  side 354 


<24 


CO-MEMS. 


kRTICLB  PAfll 

Division  of  Trapezoids. 

(544)  By  lines  parallel  to  tne  bases 35 

(546j  By  lines  starting  from  points  in  a  side 355 

(547 j  Other  eases 356 

Division  of  Quadrilateral $. 

(54§)  By  lines  parallel  to  a  side 356 

(549)  By  lines  perpendicular  to  a  side 358 

(550)  By  lines  running  in  any  given  direction 358 

(551)  By  lines  starting  from  an  angle 353 

(552)  By  lines  starting  from  points  in  a  side 358 

(554)  By  lines  passing  through  a  point  within  the  figure 359 

Division  of  Polygons. 

(555)  By  lines  running  in  any  given  direction 360 

(556)  By  lines  starting  from  an  angle 360 

(557)  By  lines  starting  from  a  point  on  a  side 361 

(558)  By  lines  passing  through  a  point  •within  the  figure 361 

(559)  Other  Problems   361 


PART    XII. 
FMTED  STATES'  PUBLIC   LA\DS. 


(560)  General  system 363 

(561)  Difficulty 364 

(562)  Running  township  lines. . . .  366 

(563)  Running  section  lines 368 

(564)  Exceptional  methods 370 

Water  fronts 370 

Geodetic  method 371 


Meandering 371 

(565)  Marking  lines 372 

(566)  Marking  cotners 372 

(567)  Field-books 376 

Township  lines 377 

Section  lines 878 

Meandering 373 


APPENDIX 


APPENDIX  A.    Synopsis  of  Plane  Trigonometry. 


(1)  Definition 379 

(2)  Angles  and  Arcs 379 

(3)  Trigonometrical  lines 380 

(4)  The  lines  as  ratios 381 

(5)  Their  variations  in  length  ....  381 

(6)  Their  changes  of  sign 382 


(7)  Their  mutual  relations 383 

(§)  Two  arcs 383 

(9)  Double  and  half  arcs 384 

(10)  The  Tables 384 

(11)  Right-angled  triangles 385 

(12)  Oblique-angled  triangles ....   385 


APPENDIX  B.    Demonstrations  of  ProWems,  &c. 


Theory  of  Transversals 387 

Harmonic  division 390 

The  Complete  Quadrilateral 391 

Proofs   of    Problems    in   Part  II., 
Chapter  V. 393 


J 

Proofs  of  Problems  m  Part  V 397 

in  Part  VI 398 

in  Part  VII 399 

in  Part  XL 401 


APPENDIX  C.    Introduction  to  Levelling. 

(1)  The  Principles 409  |  (3)  The  Practice 411 

(2)  Tlie  Instruments 409 


TRAYEKSE  TABLES: 


OE, 


LATITUDES  AND  DEPARTURES  OF  COURSES, 


CALCULATED  TO 


THREE    DECIMAL    PLACES: 


EACH    QUAKTER    DEGREE    OF    BEARING. 


LATITUDES    AND    DEPARTURES.                                | 

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0.526 

1-701 

I  .o52 

2.55i 

1-379 

3.401 

2- jo5 

4 

252    ! 

58i 

32° 

0.848 

o.53o 

1.696 

1.060 

2.544 

I  -590 

3.392 

2.120 

4 

24o 

58° 

324 

0.846 

0.534 

I  -691 

1 .067 

2.537 

I -60 1 

3-383 

2.134 

4 

229 

^-1 

32i 

0-843 

0.537 

1-687 

I  -075 

2.53o 

I  -612 

3-374 

2.149 

4 

217 

57i 

32j 

o.84i 

0.541 

1-682 

1-082 

2.523 

1-623 

3-364 

2.164 

4 

2o5 

57i 

33° 

0.839 

0.545 

1-677 

1-089 

2.5i6 

1-634 

3-355 

2.179 

4 

193 

57° 

33i 

0.836 

0-548 

1-673 

1-097 

2.509 

1-645 

3.345 

2.193 

4 

181 

56| 

33.i 

0.834 

0-552 

1-668 

1-104 

2.502 

1-656 

3-336 

2.208 

4 

169 

56i 

33-J 

o.83i 

0-556 

1-663 

i-iii 

2.494 

1.667 

3-326 

2-222 

4 

i57 

56i 

34° 

0.829 

0.559 

1.658 

1-118 

2.487 

1-678 

3.3i6 

2-237 

4 

145 

56° 

34i 

0.827 

0.563 

1-653 

1-126 

2.480 

1-688 

3 -306 

2-25l 

4 

i33 

55J 

34i 

0.824 

0-566 

1-648 

1-133 

2.472 

1-699 

3-297 

2.266 

4 

121 

55i 

34i 

0.822 

0.570 

1-643 
1-638 

I-  i4o 

2.465 

1-710 

3.287 

2.280 

4 

108 

55i 

35° 

0.819 

0-574 

1-147 

2-457 

1-721 

3-277 

2.294 

4 

096 

55° 

35i 

0.817 

0-577 

1-633 

i.i54 

2.450 

1-731 

3-267 

2.309 

4 

o83 

54} 

35i 

o-8i4 

o-58r 

1.628 

1.161 

2-442 

1-742 

3-257 

2-323 

4 

071 

54i 

35| 

0.812 

0.584 

1.623 

i-i68 

2-435 

1-753 

3-246 

2.337 

4 

o58 

54i 

36° 

0.809 

0-588 

1.618 

1-176 

2.427 

1-763 

3-236 

2.35i 

4 

o45 

54° 

36i 

0-806 

0-501 

i.6i3 

i-i83 

2.419 

1-774 

3-226 

2.365 

4 

o32 

53J 

36i 

0-804 

0.595 

1.608 

I -190 

2.412 

1-784 

3-215 

2-379 

4 

019 

53i 

36| 

0-801 

0.598 

i.6o3 

I -197 

2.404 

1-79^ 

3 -205 

2.393 

4 

006 

53i 

37° 

0-799 

0.602 

1.597 

I  -204 

2.396 

i-8o5 

3.195 

2.407 

3 

993 

53° 

-^7i 

0-796 

o-6o5 

1 .592 

I  -211 

2.388 

1-816 

3  i84 

2.421 

i 

980 

b2| 

3?^ 

0-793 

0-609 

1.587 

1-218 

2.38o 

1-826 

3-173 

2.435 

3 

967 

52i 

37l 

0-791 

0.612 

i.58i 

I  -224 

2.372 

1-837 

3-i63 

2-449 

3 

953 

t)2i 

38° 

0.788 

0-616 

1.576 

1-231 

2-364 

1-847 

3-i52 

2-463 

3 

940  , 

52° 

38i 

0.785 

0-619 

1.D71 

1-238 

2-356 

1-857 

3-i4i 

2.476 

3 

927  1 

5i| 

38  i 

0.783 

0-623 

1.565 

1-245 

2.348 

1-868 

3.i3o 

2.490 

3 

9i3  : 

5ii 

38| 

0-780 

0.626 

i.56o 

I  -252 

2.340 

1-878 

3- 120 

2.5o4 

3 

899 

5ii 

39° 

0-777 

0.629 

1.554 

1-259 

2-331 

1-888 

3-109 

2.517 

3 

886  1 

51° 

39i 

0-774 

0.633 

1-549 

1-265 

2-323 

1-898 

3-098 

2.531 

3 

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5o| 

39i 

0-772 

0-636 

1-543 

1-272 

2-315 

1-908 

3-086 

2-544 

3 

858  1 

5oi 

39| 
40° 

0-769 
0-766 

0-639 

1-538 

1-279 

2 -307 

1-918 

3-075 

2-558 

3 

•844 

5oi 

0.643 

1-532 

1-286 

2.298 

1.928 

3-064 

2.571 

3 

83o 

50° 

4oi 

0.763 

0-646 

1.526 

I  -292 

2.290 

1.938 

3-o53 

2.584 

3 

hi6 

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40i 

0-760 

0-649 

I -521 

1-299 

2.281 

1-948 

3-042 

2.598 

3 

802 

49i 

4oJ 

0.758 

0.653 

i.5i5 

i-3o6 

2  273 

1-958 

3-o3o 

2. 611 

3 

788 

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41° 

0.755 

0.656 

1 .509 

I.3l2 

2  264 

1.968 

3-019 

2.624 

3 

774 

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4ii 

0.752 

0-659 

i.5o4 

1-319 

2.256 

1.978 

3-007 

2.637 

3 

739 

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4ii 

0.749 

0.663 

1.498 

1-325 

2.247 

1.988 

2.996 

2.65o 

3 

745 

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0-746 

0.666 

1.492 

1-332 

2.238 

1.998 

2.984 

2.664 

3 

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48  i 

48° 

0-743 

0-669 

1.486 

1-338 

2.229 

2.007 

2-973 

2-677 

3 

7.6 

48° 

42i 

0-740 

0-672 

1.480 

1-345 

2.221 

2.017 

2-901 

2.689 

3 

701 

47| 

42i 

0.737 

0-676 

1-475 

1-351 

2.212 

2.027 

2-949 

2.702 

3 

686 

474 

42i 

0.734 

0-679 

1-469 

1-358 

2.2o3 

2.036 

2-937 

2.715 

3 

672 

47k 

43° 

0.731 

0-682 

1-463 

1-364 

2.194 

2 .  046. 

2-925 

2.728 

3 

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47° 

43i 

0.728 

0.685 

1.457 

1-370 

2.185 

2.o56 

2-913 

2.741 

3 

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4<H 

43i 

0.725 

0.688 

I-45I 

1-377 

2.176 

2.o65 

2-901 

2.753 

3 

627 

46  ii 

43J 

0.722 

0.692 

1-445 

1-383 

2.167 

2 .  075. 

2-889 

2.766 

3 

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46i 

44° 

0-719 

0-695 

1-439 

1-389 

2.158 

a.o84 

2-877 

2-779 

3 

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46° 

44  i 

0.716 

0-698 

1-433 

1-396 

2-149 

2-093 

2-865 

2.791 

3 

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44i 

0.713 

0-701 

1-427 

I-402 

2  . 1 4o 

2-io3 

2-853 

2 -804 

3 

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4H 

44i 

0.710 

0-704 

1-420 

i-4o8 

2.l3l 

2-112 

2-841 

2-816 

3 

.551 

45i 

45^' 

0.707 

0-707 

i-4i4 

i-4i4 

2-I2I 

2-121 

2-828 

2-828 

3-536 

45° 

CT3 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

1 

2 

t 

3 

-i 

J 

5 

LATITUDES  AND  DEPARTURES.             } 

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c^   1  Dep. 

6 

■7       ! 

8 

0 

tab 

Lat. 

Dep.  1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

30° 

3ot 
3oi 
3n| 
31° 
3.^ 
3ii 
3.. if 
32° 

32.t 
32  i 

32  J 

33° 

33i 
33.i 

33  i 
34° 

34i 
34i 
34i 

35° 

35  i 
35i 
35| 
3«° 

36i 

36  i 
36} 
3T° 
37i 
37i 
37i 
38° 
38i 
38i 
38| 
39° 
39i 
39i 
391 

40° 

4oi 
4oi 
4o| 

41° 
4ri 
4ii 
4i| 

42° 

42i 

42  i 

42j 

43° 

43i 
43i 
431 
44° 
44i 
44i 
441 
45° 

E33 

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cf<a 

2.5oo 
2-5i9 
2-538 
2-556 
2-575 
2-594 
3-612 
2-63i 
2-65o 
2-668 
2-686 
2-705 
2-723 

2-741 

2.760 

2.778 

2.796 
2.814 

2.832 

2.850 

5- 

5- 

5- 

5- 

5. 

5 

5 

5 

5 

5 

5 

5 

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5 

4 

4 

4 

4 

4 

196 

i83 
170 
i56 
1 43 
129 
116 
102 
088 
074 
060 
o46 
o32 

018 

oo3 
989 
974 
960 
945 
930 

3- 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

000 

023 

045 
068 
090 
ii3 
i35 
1 57 
£80 
202 
224 
246 
268 
290 

3l2 

333 
355 

377 
398 
420 

6 
6- 
6 
6 
6 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

062 
047 

o3i 

016 

000 

984 

968 

952 

936 

920 

904 

887 

871 

854 

837 

820 

8o3 

786 

769 

752 

3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 

5oo 
526 
553 

579 
6o5 
63 1 
657 
683 
709 
735 
761 
787 
812 
838 
864 
889 
914 
940 
965 
990 

6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 

928 
9ti 
893 
875 
857 
839 
821 
8o3 
784 
766 
747 
728 
709 
690 
671 
652 
632 
6i3 
593 
573 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

000 
o3o 
060 
090 
120 
i5o 
80 
210 
239 
260 
298 
328 
357 
386 
4i6 
445 
474 

5o2 

53i 

56o 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 

-7 

7 
7 

794 
775 
755 
735 
7i5 
694 
674 
653 
632 
612 
591 
569 
548 
527 
5o5 
483 
46i 
439 
417 
395 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
5 
5 
5 
5 
5 

5oo 
534 
568 
602 
635 
669 
702 
736 
769 
802 
836 
869 
902 

967 
000 
o33 
o65 
098 
i3o 

60° 

59J 
59i 

59i 
59° 

58 1 
58i 
58  i 
58° 
57J 
57i 
57i 
57° 
56| 
56i 
56i 
56° 
55| 
55i 
55i 

2.868 
2.886 
2 .  904 
2.921 
2.939 
2.957 
2-974 
2-992 
3-009 
3-026 
3-044 
3-061 
3-078 
3-095 
3-ii3 
3-i3o 
3-147 
3-164 
3-i8o 
3-197 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

915 

900 
885 
869 
854 
839 
823 
808 
792 
776 
760 
744 
728 
712 
696 
679 
663 
646 
63o 
6i3 

3 
3 
3 
3 
3 
3- 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 

441 
463 
484 
5o5 
527 
548 
569 
590 
611 
632 
653 
673 
694 
7i5 
735 
756 

776 
796 
816 
.837 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

734 
716 
699 
681 
663 
645 
627 
609 
590 
572 
554 
535 
5i6 
497 
478 
459 
440 
421 
4oi 
382 

4 
4 
4 
4 
.4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

oi5 
o4o 
o65 
090 
ii5 
139 
1 64 
188 

2l3 

287 
261 
286 
3io 
334 
358 
38 1 
4o5 
429 
453 
476 

6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 

553 
533 
5i3 
493 
472 
452 
43i 
4io 
389 
368 

347 
326 
3o4 
283 
261 
239 
217 
,95 
173 
i5i 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
5 
5 
5 
5 
5 

589 
617 
646 
674 
702 
73o 
759 
787 
8i5 
842 
870 
898 
925 
953 
980 
007 
o35 
062 
089 
116 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
6 
6 
6 
6 

372 
35o 
327 
3o4 
281 
258 
235 
211 
188 
1 64 
1 40 
116 
092 
068 
043 
019 

994 
970 
945 
920 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

162 
194 
226 
258 
290 

322 

353 
385 
4i6 
448 

479 
5io 

54i 
572 
6o3 
633 
664 
694 
725 
755 

55° 
541 
54  i 
54i 
54° 
53J 
53i 
53i 
53° 

52j, 

52i 

52i 

52° 

5.| 
5ii 
5ii 
51° 

5o| 
5oi 
5o;l 

3-214 
3.23i 
3-247 
3.264 
3-280 
3-297 
3-3i3 
3-329 
3-346 
3-362 
3-378 
3.394 
3-4io 
1  3-426 
3-442 
3-458 
3-473 
3-489 
3.5o5 
3-520 
3-536 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

596 
579 
562 
545 
528 
5ii 
494 
•476 
•459 
•44i 
-424 
•406 
.388 
.370 
.352 
.334 
.316 
.298 
.280 
•  261 
-243 

3 
3 
3 
3 
3 
3 
3 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

.857 

•877 
.897 

.917 
.936 
-956 
-976 
-995 
.oi5 
-o34 
-o54 
.073 
-092 
-III 
-i3o 
-149 

•  168 
.187 

•  206 
-224 
-243 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
4 
4 
4 

362 
343 
323 
-3o3 
-283 
-263 
-243 
-222 
-202 

•  182 
.161 

•  i4o 
-119 
-099 

•  078 
•o57 
.035 
.014 
-993 
-971 

•  950 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

5  00 
523 
546 
569 
592 
6x5 
638 
66  r 
-684 
-707 
•  729 
752 
•774 
-796 
.818 
84i 
863 
885 
-906 
-928 
-950 

6 
6 
6 
6 
6 
6 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

128 
106 
08  3 
061 
o38 
oi5 

968 
945 
922 
898 
875 
85i 
827 
8o3 

779 
755 
780 
706 
681 
657 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

142 
.69 
196 
222 
248 
275 
3oi 

327 

353 
379 
4o5 
43o 
456 
48 1 
5o7 
532 
557 
582 
607 
632 
657 

6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 

894 
869 
844 
818 
792 
767 
74i 
7i5 
688 
662 
635 
609 
582 
555 
528 
5oi 
474 
447 
419 
392 
364 

5 
5 
5 
5 
5 
5 
5 
5 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 

785 
8i5 
845 
875 
905 
934 
964 
993 
022 
o5i 
080 
109 
1 38 
167 
.95 
224 

252 
280 

3o8 
336 

364 

1 

50° 

49J 
49i 
49i 
49° 
48| 
484 
48i 
48° 
47* 
47i 
47i 

470 

46| 
46i 
46i 
46* 

45J 
45i 

4Si 
45° 

tib 

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Lat. 

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Lat. 

Dep. 

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6 

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8 

0 

TABLE  OF  CHORDS:  [Radius —  I.OOOOJ.          | 

M. 

o' 

I 
2 

3 

4 
5 
6 
7 
8 

9 

lO 

11 

12  1 

i3 
i4 
i5 
i6 

17 
i8 
19 
20 

21 

;22 
23 

25 

26 

27 

28 

3o 
3i 

32  1 

33; 

34 
35 
36 

37 
38 
39 
4o 

4i 
4a 
43 

45 
46 

47 
48 

t 

5i 

52 

53 
54 
55 
56 

57 
■58 

? 

0° 

1° 

2° 

3° 

4° 

5° 

.0872 
.0875 
.0878 
.0881 
.0884 
.0887 
.0890 
.0893 
.0896 
.0899 
.0901 

6° 

yo 

8° 

9° 

10° 

u. 

0' 
I 
a 
3 
4 
5 
6 

7 
8 

9 

10 

II 
12 
i3 
i4 
i5 
16 

17 
18 

19- 

20 

21 

S2 

23 

24 

25 

26 

27 
28 

=9 
3o 

3i 

32 

33 
34 
35 
36 

37 
38 

39 
4o 

4i 
42 
43 
44 
45 
46 
47 
48 

n 

5i 

52 

53 

54 
55 
56 
57 
58 

60 

.0000 
.ooo3 
.0006 

•  0009 

•  0012 

•  ooi5 

•  0017 

•  0020 

•0023 

•  0026 

•  0029 

•  0175 

.0177 
.0180 
.oi83 
.0186 
.0189 
.0192 
.0195 

•  0198 

•  0201 

•0204 

.0349 
.o352 
.o355 
.0358 
.o36i 

•  0364 
.o366 
.0369 
.0372 

•  0375 
.0378 

.o524 

•  0526 

•  0529 
.o532 
.0535 
.o538 
.o54i 
.o544 
■o547 
.o55o 

•  o553 

.0698 
.0701 
.0704 
.0707 
.0710 
.0713 

•  0715 
.0718 

•  0721 
.0724 
.0727 

•io47 
.io5o 
.io53 
.1055 
.io58 
.1061 
.1064 

•  1067 
.1070 

•  IC73 
.1076 

•  I22I 

•  1224 
.1227 
.I23o 

•  1233 

•  1235 
.1238 
.1241 
■  1244 

.1247 
•I25o 

•  1395 
.1398 
.i4oi 
.i4o4 
•1407 

•  i4io 

•  i4i3 
.i4i5 
.i4i8 
.1421 
•1424 

•  i569 

•  1572 

•  1575 

•  1578 

•  i58i 
.1584 

•  I  587 

•  1589 

•  1 592 
.1595 
.1598 

•1743 
•1746 
.1749 

•  1752 

•  1755 

•  1758 
.1761 
.1763 
.1766 
.1769 
.1772 

•  0032 

•  oo35 

•  oo38 
.oo4i 

•  oo44 

•  0047 

•  0049 

•  oo52 

•  oo55 
.oo58 

.0207 

•  0209 

•  0212 
.02l5 
.0218 
.0221 
.0224 

•  0227 

•  023o 

•0233 

•  o38i 
.o384 
.0387 

•  0390 
.0393 

•  0396 

•  0398 
.o4oi 
.o4o4 
.0407 

.0556 

•  o558 

•  g56i 

•  0564 
.0567 
.0570 
.0573 
.0576 

•  0579 
.o582 

•  0730 
.0733 
.0736 
.0739 

•  0742 

•  0745 

•  0747 

•  0750 

•  0753 

•  0756 

.0904 
.0907 
.0910 
.0913 
.0916 

•  0919 
.0922 
.0925 

•  0928 

•  0931 

.1079 
.1082 
.1084 
.1087 
.1090 
.1093 
.1096 
.1099 
.1102 
•  iio5 

•  1253 

•  1256 

•  1259 

•  1262 

•  1265 
.1267 

•  1270 

•  1273 

•  1276 

•  1279 

.1427 
.i43o 
.1433 
.1436 
.1439 
.1442 
•I  444 
•1447 
•  i45o 
.1453 

.1601 

•  i6o4 
.  1607 
.1610 
.i6i3 
.1616 
.1618 

•  1621 
.1624 
.1627 

.1775 
.1778 
.1781 
•1784 
.1787 
.1789 
.1792 
.1795 
.,798 
.1801 

.0061 

•  0064 

•  0067 
.0070 
.0073 

•  0076 

•  0079 

•  0081 

•  0084 

•  0087 

.0236 
.0239 
.0241 
•  0244 
•0247 

•  025o 

.0253 

•  0256 
.0259 
.0262 

.o4io 
.o4i3 
.o4i6 
.0419 
.0422 
.0425 
.0428 
.o43o 

•  0433 

•  0436 

.o585 

•  od88 

•  0590 

•  0593 
.0596 
.0599 
.0602 

•  o6o5 

•  0608 

•  0611 

.0759 

•  0762 

•  0765 
.0768 

•  0771 
•0774 
.0776 
.0779 
.0782 
.0785 

.0933 
.0936 
.0939 
.0942 
•0945 
.0948 
.0951 
.0954 
.0957 
•  0960 

.1108 

•  mi 

•  iii4 
.1116 
.1119 

•  1122 
.1125 

.1128 
.ii3i 

•  1134 

•  1282 

•  1285 

•  1288 

•  1291 

•  1294 

•  1296 

.r299 

.l302 

i3o5 

•  i3o8 

•  1456 
•1459 
.1462 
.1465 
.1468 
.1471 
.1473 
•1476 

.1479 
.1482 

.i63o 
.1633 
.i636 
.1639 
.1642 
.1645 
•1647 
.i65o 
.1653 
.1656 

.1804 
.1807 
.1810 
.i8i3 
.1816 
.1818 
.1821 
.1824 
.1827 
•  i83o 

.0090 
.0093 

•  0096 

•  0099 

•  0102 

•  oio5 

•  0108 

•OIII 

.oii3 

•  0116 

•  0265 

•  0268 

•  0271 
.0273 
.0276 
.0279 
.0282 
.0285 1 
.0288 
.0291 

.0439 
•0442 
.0445 
.0448 
.045 1 
.0454 
.0457 
■  o46o 
•  0462 
.o465 

■  o6i4 
.0617 
.0619 
.0622 
.0625 
.0628 
.o63i 
.0634 
.0637 
•  o64o 

•  07S8 

•  0791 
•0794 

•  0797 

•  0800 

•  o8o3 
.c8o6 
.0808 
.0811 
.0814 

.0962 
.0965 
.0968 
.0971 
.0974 
.0977 
.0980 
.0983 
.0986 
.0989 

.ii37 
.ii4o 
.1143 
.1145 
.1148 
.ii5i 
.1154 
.1157 
.1160 
.ii63 

.i3ii 

•  i3i4 

•  i3i7 

•  l320 

•  i323 

•  i325 

•  i328 
.i33i 
.1334 
.1337 

•  i485 

•  I  488 

•  1491 

•  1494 
.1497 

•  i5oo 

•  l502 

.i5o5 
.i5o8 
.i5ii 

.1659 
.  1662 
.i665 

•  I  668 

•  1671 

•  1674 

•  1676 
.1679 

•  1682 

•  I  685 

.i833 
.1836 
.1839 
.1842 
.1845 
•1847 
.i85o 
.i853 
.j856 
.1859 

.0119 
.0122 

•OI25 

.0128 

•  oi3i 

•  oi34 

•  oi37 

•  oi4o 
.0143 

•  0145 

.0294 
.0297 
.o3oo 

•  o3o3 
.o3o5 
.o3o8 

•  o3ii 

•  o3i4 
.o3i7 

•  0320 

.o463 

•0471 

.0474 

•0477 

.0480 

.04831 

.0486 

.0489 

.0492 

•0494 

.0643 
.0646 
.0649 
.065 1 
.0654 
.0657 
.0660 
.o663 
.0666 
•  0669 

.0817 
.0820 
.0823 
.0826 
.0829 
.0832 
.0835 
.0838 
.o84c 
•  0843 

•  0992 
•0994 
.0997 
.1000 
.ioo3 
.1006 
.1009 
.1012 

•  ioi5 
.1018 

.1166 
•  1 169 
.1172 
.1175 
.1177 
.1180 
.1183 
.1186 
.1189 
.1192 

.i34o 
.1343 
.1346 
.1349 
.i352 
.i355 
.1357 
.i36o 
.i363 
.i366 

.i5i4 
.i5i7 

.l520 

.i523 
.1526 
.1529 
.i53i 
.i534 
.1537 
•  i54o 

.1688 

•  1691 

•  1694 

•  1697 

•  1700 
.1703 

•  1705 
.1708 
.1711 
•1714 

.1862 
.1865 
.1868 
.1871 
.1873 
.1876 
.1879 
.1882 
.i885 
.1888 

.oi48 

•  oi5i 

•  0154 

•  oi57 

•  0160 

•  oi63 

•  0166 
.0169 

•  0172 

•  0176 

.o323 

•  o326 
.0329 
.o332 
.o335 
.0337 

•  o34o 

•  o343 
c346 

.0349 

•0497 
.o5oo 
.o5o3 

•  o5o6 

•  o5o9 

•05l2 

.o5i5 
.o5i8 

•052I 

.o524 

.0672 
.0675 
.0678 
.0681 
.o683 
.0686 

•  0689 

•  0692 

•  0695 

•  0698 

.0846 
.0849 
.o852 
.0855 
.o858 
.0861 
.0864 

•  0867 
.0869 

•  0872 

.1021 

•1023 

•  1026 

•  1029 

.I032 

.1035 
.io38 

•  io4i 
.1044 
•1047 

.1195 
.1198 
.1201 
.1204 
.1206 
.1209 

•  1212 

.I2l5 

•  1218 

•  1221 

•  1369 
.1372 
.1375 

•  1378 

•  i38i 

•  I  384 

•  i3S6 

•  I  389 

•  1392 

•  [395 

•  I  543 

•  1546 

•  I  549 

•  i552 

•  i555 
.1558 

•  i56o 
.i563 

•  I  566 
.1569 

.1717 
.1720 
.1723 
.1726 
•  1729 
.1732 
•1734 
•1737 
•1740 
•1743 

.1891 
.1894 
.1897 
.1900 
.1902 

•  1905 

•  1908 

•  1911 
•1914 

•  1917 

TABLE  OF  CHORDS:  [Radius  =  1.0U0i»].           j 

M. 

o' 

1 

2 

3 

4 
5 
6 

7 
8 

9 

lO 

II 

12 
i  >3 

i4 
|i5 
ii6 

1  ' '' 

20 
21 

26 
27 
28 

3c 

3i 

32 

33 
34 
35 
36 

37 
38 
39 
4c 

4i 
42 
43 
44 
45 
46 
4- 
48 

49 
5o 

11° 

12° 

13° 

14° 

•2437 
.  2440 
.2443 
.2446 
•2449 
.2452 
.2455 
.2458 
.2460 
.2463 
.2466 

1 
15° 

16° 

17° 

18° 

.3129 
.3(32 
.3i34 
.3i37 
.3i4o 
.3i43 
.3x46 
.3x49 
.3i52 
.3i55 
•  3x57 

19° 

20° 

21° 

H. 

0 

I 
2 

3 
4 
5 
6 

7 
8 

9 
10 

XX 

12 
i3 
i4 
x5 
x6 

\l 

19 
20 

2X 

22 
23 
24 
25 
26 

28 

3x 

32 

33 
34 
35 
36 

37 

39 
40 

4i 

42 
43 
44 
45 
46 

47 
48 
49 
5o 

5x 

52 

■53 
54 
55 
56 

57 
58 

59 
60 

.1917 
.  1 920 
.1923 
.1926 
.1928 
.1931 
.1934 
.1937 
.1940 
•1943 
•  1946 

.2091 
.2093 

•  2096 

•  2099 

•  2102 
.2io5 

.2108 

.2111 
•2Il4 
•2II7 
.2119 

.2264 
.2267 
.2270 
.2273 

•  2276 

.2279 
.2281 
.2284 

.2287 
.2290 

•  2293 

.261 1 
.26x3 
.2616 
..2619 
.2622 
.2625 
.2628 
.263 1 
.2634 
.2636 
.2639 

.2783 
.2786 
.2789 
.2792 
.2795 
.2798 
.2801 
.2804 
.2807 
.2809 
.2812 

.2956 
.2959 
.  2962 

•  2965 
.2968 

•  2971 

•  2973 

•  2976 

•  2979 

•  2982 

•  2985 

.33ox 
.33o4 
.3307 
.33io 
.33x2 

•  33x5 

•  33x8 
.3321 

•  3324 
.3327 
.3330 

.3473 
•3476 

•3479 

•  3482 

•  3484 
•3487 
.3490 
•3493 
.3496 

•  3499 

•  3502 

•  3645 

•  3648 
.365o 
.3653 
.3656 
.3659 

•  3662 
.36!55 
.3668 
.3670 
.3673 

•1949 
.1952 
.1955 
.1957 
.•  1 960 
.1963 
.1966 
.1969 
.1972 
.1975 

•2122 
.2125 
.2128 
.2l3l 

.2i34 

•  2x37 

•  2i4o 
.2x43 
.2146 

•  2i48 

.2296 
.2299 

•23o2 

.23o5 

•  23o7 

•23lO 

•  23x3 

•  23i6 

•  2319 

•2322 

.2469 
.2472 
•2475 
.2478 

•  2481 

•  2484 

•  2486 
.2489 

•  2.492 

•  2495 

.2642 

.2645  ' 

.2648 

.265x 

.2654 

.2657 

.2660 

•  2662 

•  2665 
.2668 

.28x5 
.2818 
.2821 
.2824 
.2827 
.283o 

.2832 

.2835 
.2838 
.2841 

.2988 
2991 

■  2994 
•  2996 
.2999 
.3oo2 
.3oo5 
.3008 
.3oxi 
.3oi4 

.3i6o 
.3i63 
.3i66 
.3169 
.3172 
•  3x75 
•3178 
.3i8o 
3i83 
.3x86 

.3333 
.3335 
.3338 
.3341 
.3344 
•3347 

•  335o 

•  3353 

•  3355 

•  3358 

.35o4 
.3507 
.35x0 
.35x3 
.35x6 
.35x9 

.3d22 

■  3525 
.3527 
.3530 

.3676 
.3679 
.3682 
.3685 
.3688 
.3690 
.3693 
.3696 
.3699 
.3702 

.1978 
.1981 
.1983 
.1986 

•  19S9 
.1992 
.1995 
.1998 

•  2001 

•  2004 

•2l5l 

.2i54 
.2x57 
.2x60 
.2x03 
.2x66 

•  2x69 

•  2172 
•2174 
.2x77 

.2325 

.2328 

.233i 
.2333 
.2336 

.2339 
.2342 

.2345 
.2348 
.235x 

.2498 
.25oi 
.25o4 

.2507 

.25lO 
•25l2 

.25 1 5 

•  25x8 

•2521 

•  2524 

•  2671 
•2674 
•2677 
.2680 
.2683' 

•  2685 
.2688 
•2691 
•2694 

•  2697 

.2844 
.2847 
.2850 
.2853 
.2855 
.2858 
.2861 
.2864 
.2867 
.2870 

.3017 
.3019 

•3022 

•  3025 
.3028 
■  3o3i 
.3o34 
.3o37 
.3o4o 

•  3o42 

.3189 

•  3192 
.3195 
.3198 

•3200 

•  32o3 

•  32o6 

•  3209 

•3212 

•  32x5 

•  336i 

•  3364 
.3367 

•  3370 
.3373 
.3376 
.3378 
.3381 

•  3384 
.3387 

.3533 
.3536 
.3539 
.3542 
.3545 

•3547 

•  355o 

•  3553 
•3556 
.3559 

.3705 
.3708 
.37x0 
.3713 
.3716 
.3719 
.3722 
.3725 
.3728 
.3730 

.2007 

.20IO' 

.2012 

.20X5 

.2018 

.2021 

.2024 

.2027 

.2o3o 
•  2o33 

.2180 
.2x83 
.2186 
.2189 
.2192 
.2195 
.2198 
■  2200 

.2203 
•  2206 

.2354 
.2357 
.2359 

•2362 

.2365 
.2368 

•  2371 
•2374 

•  2377 

•  2380 

.2527 
.253o 
.2533 
.2536 
.2538 
.254x 
•2544 
•2547 
•  2550 
.2553 

.2556 
.2559 
.256i 
.2564 
.  2567 
.2570 
.2573 
.2576 
.2579 
.2582 

•  2700 
.2703 

•  2706 

•  2709  i 
.2711 
.2714 
.2717 
•2720 
.2723 
.2726  I 

I 

•  2873 
.2876 
.2878 
.2881 
.2884 
.2887 
.285^ 
.2893 

•  2896 

•  2899 

•  3o45 
.3o48 
.3o5i 
.3o54 
.3o57 
.3  060 
.3o63 
.3o65 
.3o68 

•  3071 

•  32x8 

•  3221 
.3223 
.3226 
.3229 
.3232 

.3235 
.3238 
.3241 
•3244 

.3390 
.3393 
.3396 
.3398 

•  34oi 
.3404 
•3407 
."34 10 
.34x3 

•  34x6 

•  3562 

•  3565 

•  3567 

•  3570 
.3573 
.3576 
.3579 
.3582 
.3585 
.3587 

.3733 
.3736 
•3739 
•3742 
•3745 
.3748 
.3750' 

•  3753 

•  3756 
•3759 

.2o36 
.2o38 
.204 1 

•  2044 
•2047 
■  2o5o 
.2o53 
.2o56 

•  2059 
.2062 

•  2205 
•2212 
.22X5 
.22X8 
.2221 
.2224 

•  2226 
.2229' 

•  2232 
•2235 

.2383 
.2385 
.2388 
.2391 
•2394 
.2397 
.2400 
.  24o3 

•  2406 

•  2409 

.2411 
•24i4 
•2417 

.24-20 

.2423 

•  2426 

•  2429 
.2432 

•  2434 
•2437 

.2729 
.2732 
•2734 
.2-37 
•2740 
•2743 

•  2746 
•2749 
.2752 

•  2755 

.2758 
2760 
.2763 
.2766 
.2769 
.2772 

•  2775 
.2778 

•  278X 

•  2783 

.2902 
.2904 
•2907 
.2910 
.29x3 
.2916 
.29x9 
.2922 
.292? 
•2927 

•3074 
.3077 
.3o8o 
.3o83 
.3o86 
.3o88 
.3091 
.3094 
.3097 
•  3x00 

.3246 
.3249 

.3252 

.3255 
.3258 
.3261 
.3264 
.3267 

•  3269 

•  3272 

•3419 
.3421 
.3424 
•3427 
.3430 
.3433 
.3436 
.3439 
•3441 
•3444 

•  3590 
.3593 
.3596 
.3599 
.3602 

•  36o5 
.3608 

•  36io 
.36x3 

•  36x6 

.3762 

.3765 

.3768 

•3770 

•3773 

•3776 

.3779' 

.3782 

.3785 

.37S8 

5i 

52 

53 
54 
55 
56 
57 
58 

i9 

60 

.2o65 
.2067 
•  2070 
.2073 
.2C76 
.2079 
.2082 
.2085 
.2088 
.2091 

.2238 
•  2241 
•2244| 

.2247 

.  225o 
.2253 
.2255 
.2258 

.2261 

.2264; 

.2585 
.2587 
.2590 
.2593 
.2596 
.2599 
.  2602 
.2605 
.2608 
■  2611 

.  2930 
.2933 

•  2936 
.2939 
.2942 
•2945 
.2948 

•  2950 

•  2953 
.2956 

.3io3 
.3 106 
.3109 

•  3ixi 

•  31x4 

•  3xx7 
.3120 
.3123 
.3126 
.3129 

.3275 
.327S 
.328x 
.3284 
.3287 
.3289 
.3292 
.3295 
.3298 
•  33oi 

•3447 
.3450 
.3453 
.3456 
.3459 
.3402 
.3464 
•3467 
•  3470 
•347M 

•  36x9 

.3622 

.3d25[ 
.3628| 
.3630 
.3633 
.3636 

•  3639 

•  3642 
3645 

.3790 

3793 
3796 

3799 
38o2 

38o5 

3So8 

38ic 

38x3 

3Si6 

TABLE  OF  CHORDS:  [Radius  =  1.000.^]. 

M. 

o' 

22° 

23° 

24° 

25° 

26° 

27° 

2§° 

29° 

30°  I 

31° 

32° 

M. 

0' 

.38i6 

.3987 

.4i58 

.4329 

.4499 

.4669 

.4838 

.5oo8 

.5176 

.5345 

.55i3 

I 

.3819 

.3990 

.4161 

•  4332 

.4502 

•  4672 

.4841 

.5oio 

•5r79 

•  5348 

.5516 

I 

2 

.3822 

.3993 

.4164 

•  4334 

.45o5 

.4675 

•4844 

.5oi3 

•  5182 

•  535o 

.5518 

2 

3 

.3825 

.3996 

.4167 

•4337 

.4508 

•4677 

•4847 

.5016 

.5i85 

.5353 

.5521 

3 

4 

•3828! 

.3999 

.4170 

•  4340 

.45x0 

.4680 

.4850 

.5019 

.5i88 

.5356 

.5524 

4 

5 

•  3830 

.4002 

.4172 

.4343 

.45i3 

.4683 

•  4853 

.5022 

•  5190 

.5359 

.5527 

5 

6 

•  3833 

.4004 

•4175 

•  4346 

.4516 

.4686! 

.4855 

.5024 

•5193  : 

•  5362 

.5530 

b 

7 

•  3836 

•  4007 

.4178 

•4349' 

.4519 

.4689 

.4858 

.5027 

.5196  ' 

•  5364 

•  5532 

7 

8 

•  3839 

.4010 

.4181 

•  4352 

.4522 

.4692 

.4861 

.  5o3o 

.5199  ' 

.5367 

.5535 

8 

9 

•  3842 

.4oi3 

.4184 

.4354 

.4525 

.4694 

.4864 

•  5o33 

.5202  : 

.5370 

•  5538 

9 

lO 

II 

•3845 
.3848 

.4016 

•4187 

•4357 

•4527 

•4697 

.4867 

•  5o36 

•  52o4 

•  5373 

.5541  1 

0 



II 

.4019 

.4190 

.4360 

.4530 

.4700 

.4869 

•  5o39 

.5207 

.5376 

•  5543 

12 

•  385o 

.4022 

.4192 

.4363 

.4533 

.4703 

.4872 

.5o4i 

.5210' 

.5378 

•  5546 

13 

i3 

•  3853 

.4024 

.4195 

.4366 

•  4536 

.4706 

•4875 

.5044 

.52i3 

.5381 

.5549 

i3 

i4 

•  3856 

•  4027 

.4198 

.4369 

.4539 

.4708 

•  4878 

•5o47 

.5216 

.5384 

•  5552 

i4 

i5 

•  3859 

•  4o3o 

•4201 

•4371 

.4542 

•4711 

•  4881 

.5o5o 

.5219 

.5387 

.5555 

i5 

|i6 

.3862 

.4033 

.4204 

•4374 

.4544 

.4714 

•  4884 

.5o53 

.5221 

•  5390 

•  5557 

lb 

17 

.3865 

•  4o36 

.4207 

•4377 

•4547 

•4717 

•  4886 

.5o55 

.5224 

.5392 

.5560 

17 

i8 

.3868 

•  4039 

.4209 

.•4380 

.4550 

.4720 

.4889 

.5o58 

.5227 

.5395 

•  5563 

lb 

19 

.3870 

•  4o42 

.4212 

.4383  1 

.4553 

•4723 

.4892 

.5061 

.5230 

.5398 

•  5566 

19 

20 
21 

.3S73 

.4044 

.42r5 

.4386  ' 

.4556 

.4725 

.4895 

.5064 

.5233, 

.5401 

.5S69 

20 
21 

.3876 

•4o47 

.4218 

.4388 

.4559 

.4728 

.4898 

.5067 

.5235 

.5404 

.5571 

22 

.3879 

.4o5o 

•4221 

•4391 

•  4561 

•473r 

.4901 

•  5070 

.5238  ' 

.5406 

.5574 

22 

23 

.3882  1 

.4o53 

.4224 

•4394 

.4564 

•4734 

.4903 

.5072 

.5241  ! 

.5409 

•5577 

23 

24 

.3885! 

.4o56 

.4226 

•  4397 

-4567 

•4737 

.4906 

•  5075 

•  5244 

•  54i2 

.5580 

24 

25 

.3888, 

.4059 

.4229 

.4400  ! 

•4570 

•4740 

.4909 

•  5078 

•5247  1 

.54 1 5 

.5583 

25 

26 

.3890 

.4061 

.4232 

•44o3  1 

•4573 

•  4742 

.4912 

.5o8i 

.5249  ' 

.5418 

•  5585 

26 

27' 

.3893 

.4064 

.4235 

.4405 

.4576 

•4745 

•4915 

•5o84 

.5252 

.5420 

.5588 

?-r 

28 

.3896 

.4067 

•  4238 

.4408 

•4578 

•4748 

.4917 

.5086 

.5255 

.5423 

.5591 

2b 

29 

.3899; 
.39021 

.4070 

•  4-241 

•44ii 

•  458i 

•475i 

.4920 

.5089 

.5258 

.542b 

•5594 

^9 

3o 
3i 

•  4073 

•4244 

•44i4 

.4584 

•4754 

.4923 

•  5092 

.5261 

•W29 

•5597 

3i 

.3905 

.4076 

•4246 

•4417^ 

•4587 

•4757 

.4926 

.5095 

-5263 

.5432 

.5599 

32 

.3908 

•4079 

.4249 

.4420  ; 

.4590 

•4759 

.4929 

.5098 

.5266 

.5434 

.56o2 

32 

33 

.3910 

.4081 

.4252 

.4422  1 

.4593 

•4762 

.4932 

.5ioo 

.5269 

•5437 

.56o5 

33 

34 

.3913 

.4084 

.4255 

.4425 

.4595 

•4765 

.4934 

.5io3 

.5272 

.5440 

.5608 

34 

35 

.3916 

.4087 

•  4258 

.4428 

.4598 

•  4768 

.4937 

.5106 

.5275 

.M43 

.56ii 

35 

36 

.3919 

.4090 

.4261 

•4431 

.4601 

•4771 

.4940 

.5109 

.5277  : 

.544b 

.S6i3 

36 

37 

.3922 

.4093 

•  4263 

•4434 

.4604 

.4773 

•4943 

.5112 

.5280 

•5448 

.5616 

37 

38 

.3925 

.4096 

•  4266 

•4437 

.4607 

.4776 

•4946 

.5ii5 

•  5283 

.54S1 

.5619 

38 

39 

.3927 

.4098 

.4269 

.4439 

.4609 

•4779 

•4948 

•  5ii7 

.5286 

•t>4t>4 

.5622 

39 

40 
4t 

.3930 

•  4ioi 

.4272 

•4442! 

.4612 

.4782 

•49^1 

•5l20 

.5289' 

.5457 

.5625 

40 
4» 

.3933 

.4ic4 

•4275 

•4445 

.46i5 

•4785 

•4954 

•  5i23 

.5291 

.5460 

.5627 

42 

3936 

.4107 

•4278 

•4448 

.4618 

.47«8 

.49^7 

•  5126 

.5294  1 

.5462 

.563o 

42 

43 

3939 

.4110 

•  4280 

•445i 

.4621 

•  4790 

.4960 

•  5129 

.5297  1' 

•  5465 

.5633 

43 

Vi 

.3942 

.4ii3 

•  4283 

.4454 

.4624 

•  4793 

.4963 

•  5i3i 

.53oo' 

•  5468 

.5636 

44 

i"" 

•3945 

.4116 

.4286 

•  4456 

.4626 

•4796 

.4965 

•  5i34 

.53o3 

•5471 

.5638 

45 

io 

.3947 

•  4118 

.4289 

•4459 

.4629 

•4799 

.4968 

•  5i37 

.53o6 

•5474 

564 1 

46 

47 

.3950 

.4121 

.42Q2 

.4462 

.4632 

•  4802 

.4971 

.5[4o 

.53o8 

.5476 

.5644 

47 

48 

.3953 

.4124 

.4295 

•  4465 

•  4635 

.4805 

•4974 

.5i43 

.53ii 

.5479 

•  5647 

48 

49 

3956 

•  4127 

.4298 

.4468 

•  4638 

.4807 

•4977 

•  5i45 

.53i4 

.5482 

.5650 

49 

5o 
5i 

.3959 

•  4i3o 

•  43oo 
.43o3 

•4-471 

•  4641 

•  4810 

•4979 
.4982 

•  5i48 

.53x7  i 
.5320 i 

.5485 

•  5652 

5a 
5i 

.3962 

.4i33 

.4474 

•  4643 

•  48i3 

.5(5i 

•  5488 

.5655 

52 

.3965 

.4135 

.4306 

•4476 

•  4646 

.4816 

.4985 

.5i54 

•5322  ! 

•  5490 

.5658 

52 

53 

.3967 

.4i38 

.4309 

•4479 

•4649 

.4819 

•  49S8 

.5i57 

.5325 

.5493 

.5661 

53 

54 

.3970 

•4i4i 

.4312 

.4482 

•  4652 

.4822 

•4991 

.5160 

.5328 

.549b 

.5664 

54 

55 

.3973 

.4:44'-43i5| 

•  4485 

•  4655 

.4824 

•4994 

.5162 

.5331 

•5499 

.5666 

55 

5rt 

.3976 

•4i47 

.4317 

•  4488 

•  4658 

.4827 

•4996 

.5i6b 

.5334  ^ 

.5502 

.5669 

5b 

57 

.3979 

•  4i5o 

.4320 

•4491 

.4660 

.4830 

•4999 

•  5i68 

•  5336 

.55o4 

.5672 

57 

58 

.3q82 

.4i53 

.43231 

-4493 

•  4663 

•  4833 

.  50O2 

•  517. 

•  5339 

.5507 

.5675 

58 

III 

.3985 

.4155 

•4326 

•4496 

.4666 

•  4836 

.5oo5 

•5t74 

•5342  , 

.55.0 

.5678 

59 

.3987 

•  4i58 

•4329! 

•4499 

.4669 

•  4838 

.5008 

•51761.5345  ' 

55i3l 

.5680 

60 

lu 


TABLE  OF  CHORDS:  LI^ai"l-s  =  1.ooOuJ.          | 

-■1 
- 

o' 

I  ! 

2 

3  ; 
4 

51 

6 

9 

I  I 

(6! 

j8| 
'9| 

20 

1 

21 
L24 

h5 
I26 

27 
28 
29 
3o 

3i  ■ 

32  ' 

33  ! 

34; 

35 

36 

37 
38 
39 

40 

4t  ' 

42 

43 

44 

45 

46 

47 

48 

49 

5o, 

5i  1 

32  ■ 

53 

'4 

55 

56: 

58  t 

59  ! 
60 

33° 

-  568o 
.5683 
.5686 
.5689 
.5691 
-5694 
.5697 
.5700 
-5703 
-5705 
•  5708 

.5711 

.5714 
.5717 
.5719 
.5722 
.5725 
-5728 
.5730 
.5733 
-5736 

34° 

35° 

36° 

37° 

-6346 
-6349 
-6352 
-6354 
-6357 
-636o 
-6363 

•  6365 

•  6368 

•  6371 

•  6374 

3§° 

39° 

40° 

41° 

42°  43° 

M. 

0 
I 
2 

3 
4 
5 
6 

0 

9 

10 



1 1 
1  i 
i3 
i4 
i5 
16 
17 
18 

•9 
20 

21 
22 

23 

24 

25 

26 

27! 

28 

29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 
4o 

4i 
42 
43 
44 
45 

t 
% 

5o 
5i 

52 

53 

54 
55 
56 

57 
58 
59 
60 

•5847 
-585o 
-5853 
-5856 
-5859 

•  5861 
-5864 

•  5867 
.5870 
-5872 
.5875 

.6014 
.6017 
.6020 
.6022 

•  6025  ■ 
.6028 

•  6a3i 

•  6o34 

•  6o36 

•  6039 

•  6042 

.6180 
.6i83 
.6186 
.6189 
-6191 
-6194 
-6197 
-6200 
-6202 
.6205 
.6208 

.65m 
.65i4 
•  65i7 
.6520 

.6522 

.6525 
.6528 
.6531 
.6533 
.6536 
.6539 

.6676 

.6679 
.6682 

•  6684 

•  6687 

•  6690 

•  6693 
.6695 
.6698 
.6701 
.6704 

-684o 

.6843 

.6846 

-6849 

-685i 

.6354  1 

.6857 

.6860  , 

.6862 

-6865 

•  6868 

•  7004 

•  7007 

•  7010 

•  7012 

•  7015 
.7018 

•  7020 
.7023 

•  7026 

•  7029 

•  7o3i 

.7167 

•  7170 
-7'73 
-7<76 

7178 
.7181 

•  7184 

•  7186 

•  7189 
.7192 
•7195 

•  7330 

•  7333 
•7335 

•  7338 
-7341 
.7344 

•  7346 

•  7349 
•73'j2 

•7354 

•  7357 

.5878 

•  5881 

•  5884 

•  5886 
.58S9 
.5S92 
.5895 
-5897 
•5900 
•5903 

.G045 

•  6047 

•  6o5o 

•  6o53 

•  6o56 
.6o58 

•  6061 

•  6064 
.6067 

•  6070 

.6211 
■  6214 

•  6216 
.6219 

•  6222 
-6225 

•  6227 

•  623o 
.6233 

•  6236 

•  6238 
.6241 
.6244 

•  6247 
.6249 

•  6252 

•  6255 

•  6258 

•  6260 
.6263 

.6266 

•  6269 
.6272 
-6274 
•6277 
-6280 

•  6283 

•  6285 

•  6288 

•  6291 

.6376 

-6379 
.6382 
.6385 
.6387 

•  6390 
.6393 

•  6396 

•  6398 

•  6401 

•  64o4 

•  6407 
.64io 

•  6412 
.64i5 

•  64i8 

•  6421 
■  6423 
-6426 

•  6429 

.6542 
.6544 
-6547 
.655o 
.6553 

•  6555 

•  6558 

•  656 1 

•  6564 

•  G566 

.6706 
.6709 
.6712 
.6715 
.6717 
.6720 

•  6723 

•  6725- 
.6728 

•  6731 

-6S70 
-6873  j 
.6876 
-6879 
.6881 

•  6884 

•  6887 

•  6890 

•  6892 

•  6895 

-7034 

•  7037 

•  7040 

•  7042 
•7045 

•  7048 
•7o5o 
•7053 
•7o56 

•  7059 

•  7197 

•  7200 

•  72o3 
■  7205 
.7208 

•  72M 
•7214 
-7216 
.7219 

•  7222 

•  7360 

•  7362 

•  7365 

•  7368 

•  7371 
-7373 
-7376 

•  7379 
.7381 
•7384 

-5739 
-5742 
.5744 
•5747 
.5750 
.5753 
•  5756 
-5758 
.5761 
•5764 

.5906-6072 
-5909  -6075 
-5911  -6078 
.5914-6081 
.5917  .6083  1 
.5920  -6086 
.5922  .6089 
•5925  ^6092 
•5928-6095 
•5931  -6097 

•  6569 

•  6572 

•  6575 

•  6577 

•  6530 

•  6583 
.6586 
.6588 
.6591 
.6594 

•6734 
.6736 

-6739 
•  6742 
-6745 
.6747 
•6750 
.6753 
•6756 
•6758 

•  6898 
.6901 
.6903  j 

•  6906 
.6909 
•6911  i 
-6914 
.6917 
.6920 

6922 

•  7061 
•.7064 

•  7067 

•  7069 
.7072 
.7075 
.7078 
.7080 
.7083 
.7086 

•  7224 
.7227 

•  723o 

•  7232 
.7235 

•  7238 
.7241 
-7243 

•  7246 

•  7249 

•  7387 
-7390 

•  73-92 
-7395 
.7398 

•  7400 

•  74o3 

•  7406 
•7408 
.7411 

•5767 
.5769 
.5772 
-5775 
•5776 
.5781 
.5783 
.5786 
.5789 
.5792 

•5934-6100  ' 
.5936  ^6103  ■ 
•5939  •6106 
.5942  •6108  1 
.5945'.6iii 
•5947-6114 
.5950-6117 
-5953  .6119 
-5956,-6122 
•5959-6125 

-6432 
.6434 
•6437 
.6440 
-6443 
.6445 
.6448 
.6451 
.6454 
-6456 

•6597 
•  6599 
.  6602 
.66o5 
.6608 
.6610 
.66i3 
.6616 
.6619 
.6621 

-6761 
-6764 
-6767 

•  6769 

•  6772 
■6775 
•6777 
.6780 
.6783 
.6786 

.6925 
.6928 
.6^31 
.6933 
.6936 
.6939 
.6941 
-6944 

•  6947 

•  6930 

.7089 
.7091 
.7094 
.7097 
.7099 
.7102 

•  7io5 
.710S 

•  7110 

•  7ii3 

•725i  .7414 
•7254  ^7417 
•7257  .7419 
•7260  ^7422 
•7262  ^7425 
•7265  .7427 
-7268  -7430 
•7270  .7433 
•7273. ^7435 
•7276  -7438 

.5795 

•5797 
.5800 
.58o3 
.5806 
.  58o8 
.58m 
.58i4 
-58i7 
•  582- 

-5961I-6128 
-5964-6130 
.5967: -61 33 
.5970  .6i36 
.5972  .6139 
.5975  ^6142 

•  5978^6144 
•5981  .6147 

•  5984  UeiSo 

.5986  ^6153  : 

.6294 
.6296 
-6299 
-63o2 
-63o5 

•  63o7 

•  63io 
.63i3 
■  63i6 

•  63i8 

-6459 
.6462 
.6465 
.6467 
.6470 
•6473 
.6476 
.6478 
.6481 
•  6484 

.6624 
.6627 

•  663o 

•  6632 
-6635 
-6638 
-664o 
-6643 
-6646 
-  6649 

.6788 
.6791 
.6794 
-6797 
.6799 
.6802 
.68o5 
.6808 
.6810 
.68i3 

.6816 
.6819 
.6821 
.6824 
.6827 
.6829 
.6832 
.6835 
.6838 
•  684o 

•  6952 
.6955 
.6958 
.6961 
.6963 
.6966 
.6969 
-6971 
-6974 
-6977 

•  7m6 

•  7m8 

•  7121 

•  7124 

•  7127 
.7129 

•  7i32 

•  7i35 

•  7137 

•  7j4o 

•7143 

•  7146 
•7148 
.7.5. 
•7154 
.7156 
.7159 
.7162 
.7165 
.7167 

-7279 

•  7281 

-7284 
.7287 
-7289 
-7292 
-7295 
.7298 

•  73oo 

•  73o3 

-7441 
-7443 
•7446 
•7449 
•7452 
•7454 
•7457 
•  7460 
.7462 
-7465 

.5822 
-5825 
-5828 
-5831 
-5834 
•  5836 
-5839 
•5842 
•5845 
•5847 

•  5985 
.5992 

•  5995 

•  5997 

•  6000 
.6oo3 

•  6006 
.6009 

•  6nii 

•  60.4 

•  6i-55  i! 
.6i58 

•  6161 
.6164 

•  6166 

•  6169 

•  6172 

•  6175 

•  6178 

•  6180 

•  6321 
.6324 

•  6327 
.633o 
.6332 
-6335 
-6338 
-634i 
-6343 
-6346 

-6487 
.6489 
-6492 
-6495 
-6498 
-65oo 

•  65o3 
.65o6 

•  65o9 
-65ii 

-6651 
-6654 
-6657 

•  6660 

•  6662 

•  6665 

•  6668 

•  6671 
.6673 

•  6676 

.6980 
.6982 
.6985 
.6988 

•6991 
.6993 

.6996 

•6999 
.7001 
.  7004 

.7306 

•  7308 

•  73m 
•73-4 

•  73i6 

•  7319 

•  7322 

•  7325 
-7327 
•733o 

•  7468 
•7471 
-7473 
-7476 

-7479 
•748 1 
•7484 
-7487 
•7489 
.7492 

11 


■                       — ■ •••" '-^ — ' ^^ =- 

TABLE  OF  CHORDS:  [li 

ADIUS  =  1.0000].             1 

M. 

o' 

44° 

45° 

46° 

.7815 

47° 

4§° 

40° 

50° 

.8452 

51° 

52° 

53° 

54° 

M. 

.7492 

.7654 

•7975 

.8i35 

.8294 

.8610 

•8767 

.8924 

.9080 

0 

I 

.7495 

•  7656 

.7817 

.7978 

.8137 

.8297 

.8455 

.86x3 

.8770 

.8927 

.9082 

I 

2 

•749« 

•7659 

.7820 

.7980 

.8i4o 

.8299 

.8458 

.86x5 

.8773 

.8929 

.9085 

2 

3 

•  7600 

.7662 

.7823 

.7983 

.8143 

.83o2 

.846o 

.86x8 

.8775 

.8932 

.9088 

3 

4 

.75o3 

.7664 

.7825 

.7986 

.8x45 

.83o4 

.8463 

.8621 

.8778 

.8934 

.9090 

4 

5  ! 

•  75o6 

.7667 

.7828 

.7988 

.8148 

.83o7 

.8466 

.8623 

.8780 

.8937 

.9093 

5 

6; 

.7508 

.7670 

•  7831 

•799' 

.8i5i 

.83x0 

.8468 

.8626 

.8783 

.8940 

.9095 

b 

-7 

•  75ti 

.7672 

.7833 

•7994 

•  8i53 

.83x2 

•8471 

.8629 

.8786 

.8942 

.9098 

7 

8| 

•75i4 

.7675 

•  7836 

•7996 

.8i5b 

.83x5 

.8473 

.863x 

.8788 

.8945 

.91UX 

8 

9 

.75.6 

•7678 

.7839 

•7999 

.8159 

.83x8 

•8476 

.8634 

.8791 

•8947 

.9x03 

9 

lO 

II 

.7519 

•  7681 

.7841 

.8002 

.816] 

.8320 

•8479 

-8636 
.8639 

•8794 

.8950 

■  QioQ 

10 
II 

•  7522 

.7683 

•7844 

.8004 

•  8:64 

.8323 

.8481 

.8796 

.8953 

.9x08 

12 

.7524 

.7686 

.7847 

.8007 

.8167 

.8326 

.8484 

.8642 

.8799 

.8955 

.91x1 

12 

i3 

.7527 

.7689 

•7849 

.8010 

•  8169 

.8328 

.8487 

.8644 

.8801 

.8958 

.91.3 

i3 

i4 

.7530 

•  7691 

•  7852 

.8012 

•  8172 

.833x 

.8489 

.8647 

.8804 

.8960 

.9X 16 

1 4 

i5 

.7533 

•7694 

.7855 

.8oi5 

•  8175 

.8334 

.8492 

.865o 

.8807 

.8963 

.9x19 

i5 

t6 

.7535 

•7697 

.7857 

.8018 

•  8177 

.8336 

.8495 

.8652 

.8809 

.8966 

.9X2X 

10 

17 

•  7538 

•  7699 

.7860 

.8020 

•  8180 

•  8339 

•8497 

.8655 

.88x2 

.8968 

.9124 

17 

iS 

•7541 

.7702 

.7863 

.8023 

•  8x83 

•  8341 

.85oo 

.8657 

.88x4 

.8971 

.9126 

x8 

19 

•7543 

•  7705 

•7865 

.8026 

•  8i85 

•  8344 

.85o2 

.8660 

.8817 

•8973 

.9129 

'9 

20 
21 

•7546 

.7707 

•  7868 

.8028 

•8x88 

•8347 

.85o5 

.8663 

.8820 

.8976 

.9x32 

20 
21 

.7549 

.7710 

.7871 

.8o3i 

•  8190 

.8349 

.85o8 

.8665 

.8822 

.8979 

•9134 

'  22 

.7551 

•77<3 

.7873 

.8o34 

.8193 

.8352 

.85x0 

•  8668 

.8825 

•  898X 

•  9x37 

22 

23  i 

.7554 

•77i5 

•7876 

.8o36 

.8196 

.8355 

.85x3 

.8671 

.8828 

.8984 

.9139 

23 

!24l 

.7557 

.77.8 

.7i;79 

.8039 

•  8198 

.8357 

.85x6 

.8673 

.883o 

.8986 

.9x42 

24 

25 

26 

.7560 

.7721 

.7882 

.8042 

•  820X 

.836o 

.85x8 

.8676 

.8833 

.8989 

•9145 

25 

.7562 

.7723 

•  7884 

.8044 

•  8204 

.8363 

.8521 

.8678 

.8835 

.8992 

.9147 

26 

27 

•  7565 

•  7726 

•7887 

.8047 

.8206 

.8365 

.8523 

.8681 

.8838 

•8994 

.9150 

27 

28 

•  7568 

.7729 

.7890 

.8o5o 

.8209 

.8368 

.8526 

.8684 

•  8841 

•8997 

•  9x52 

28 

29 

.7570 

•773 1 

.7892 

.8o52 

.8212 

.8371 

■  8529 

•8686 

•  8843 

•8999 

.9x55 

29 

3o 
3i 

.7573 

.7734 

.7895 

.8o55 

.8214 

.8373 

.853x 

.8689 

•  8846 

.9002 

.9x57 

3o 
3i 

.7576 

•7737 

.7898 

.8o58 

.82x7 

.8376 

.8534 

.8692 

•  8848 

.9005 

.9160 

32 

•7578 

•7740 

.7900 

.8060 

.8220 

.8378 

.8537 

.8694 

.885x 

.9007 

.9x63 

32 

33 

.7581 

•7742 

.7903 

.8o63 

.8222 

.8381 

.8539 

.8697 

.8854 

.9010 

.9165 

33 

34 

•7584 

•7745 

.7906 

.8066 

.8225 

.8384 

.8542 

.8699 

.8856 

•  90x2 

.9168 

34 

35 

•  7586 

•7748 

.7908 

.8068 

.8228 

.8386 

.8545 

.8702 

.8859 

.9015 

.9x70 

35 

36 

.7589 

.7750 

•79" 

.8071 

.8230 

.8389 

•8547 

.8705 

.886x 

•  90x8 

.9x73 

36 

37 

.7592 

•7753 

.7914 

.8074 

.8233 

.8392 

.8550 

.8707 

.8864 

.9020 

•9176 

37 

38 

•7595 

•7756 

.7916 

.8076 

•  8236 

.8394 

.8552 

.8710 

.8867 

.9023 

•9x78 

38 

39 

.7597 

•7758 

.7919 

•  8079 

•  8238 

.8397 

.8555 

.8712 

.8869 

.9025 

•9x81 

39 

40 

4r 

.7600 

.7761 

.7922 

.8082 

•  8241 

.8400 

.8558 

.87x5 

.8S72 

•  9028 

•  9x83 

40 
4i 

.7603 

•7764 

.7924 

.8084 

.8244 

.8402 

.856o 

.87x8 

.8874 

.9031 

.9x86 

42 

.7605 

.7766 

.7927 

.8087 

.8246 

.84o5 

.8563 

.8720 

.8877 

.9033 

.9x88  ' 

42 

4i 

.7608 

•7769 

.7930 

•  8090 

.8249 

.84o8 

.8566 

.8723 

.8880 

.9036 

•  9x91 

43 

44 

•  7611 

•  7772 

.7932 

•  8092 

.8251 

.84x0 

.8568 

.8726 

.8882 

.9038 

.9194 

44 

45 

.7613 

•7774 

.7935 

•  8095 

.8254 

.84x3 

.8571 

.8728 

.8885 

.9041 

•  9x96, 

45 

46 

•7616 

•7777 

.7938 

•  809S 

.8257 

.84i5 

.8573 

.8731 

.8887 

•9044 

•9199! 

46 

47 

.7619 

7780 

.7940 

•  8100 

.8259 

.8418 

.8576 

.8734 

.8890 

.9046 

•  920X 

47 

48 

•  7621 

.7782 

.7943 

•8io3 

.8262 

.8421 

•8579 

.8736 

.8893 

.9049 

•  9204 

48 

49 

.7624 

•  7785 

•7946 

•8io5 

.8265 

.8423 

.8581 

.8739 

.8895 

.9051 

.9207  i 

49 

5o 
5 1 

.7627 

•  7-88 

•7948 
.7951 

•8108 

.8267 
.8270 

.8426 

.8584 

•8741 

.8898 

.9054 

.9209  ; 

5o 
5i 

.7629 

•7791 

.8iii 

.8429 

.8587 

•8744 

.8900 

.9056 

.92x2 

52 

.7632 

.7793 

.7954 

.8ii3 

.8273 

.843x 

•  8589 

•8747 

.8903 

.9059 

.9214 

52 

53 

.7635 

.7796 

•7956 

•  8116 

•  8275 

.8434 

.8592 

•8749 

.8906 

.9062 

•9217  , 

53 

54 

•  7638 

•7799- 

•7959 

8119 

.8278 

.8437 

•8594 

.8752 

.8908 

.9064 

.92x9' 

54 

55 

•  7640 

.7801 

.7962 

.8121 

.8281 

.8439 

•8597 

•8754 

.89x1 

.9067 

•9222 

55 

B56 

•7643 

.7804 

•7964 

.8124 

•  8283 

.8442 

.8600 

•8757 

.89x4 

.9069 

•9225 

56 

1  5y 

.7646 

.7807 

.7967 

•  8127 

•  8286 

•8444 

.8602 

.8760 

.8916 

.9072 

•9227 

57 

158 

.7648 

.7809 

.7970 

•  8129 

.8289 

•8447 

.86o5 

.8762 

.8919 

.9075 

•9230 

58 

|59 

■7651 

.7812 

.7972 

•  8i32 

•  8291 

•  8450 

.8608 

.8765 

.8921 

.9077 

•9232 

6u  1 

i6o 

•7654 

.7815 

•7973 

•  8i35 

•  8294 

•  8452 

.86x0 

•8767 

.8924 

.9080 

•9235 

i-2 


TABLE  OF  CHORDS 

:  [Radius  =  1.0UUU]. 

1 

1  o' 

55° 

56° 

.9389 

57° 

48° 

59° 

60° 

61° 

62° 

63° 

61° 

M. 

0' 

.9235 

•9543 

•  9696 

.9848 

I. 0000 

i.oi5i 

£.o3oi 

I .0450 

£  .0598 

1   I 

.9238 

.9392 

.9546 

.9699 

•  985 1 

I . ooo3 

I.0153 

£ .o3o3 

1.0452 

£ .c6oi 

£ 

H  ^ 

.  9240 

.9395 

.9548 

.9701 

.9854 

I -0005 

i.oi56 

£ .o3o6 

£.0455 

£  .o6o3 

2 

\     '^ 

.9243 

.9397 

.9331 

•9704 

•  9856 

I . 0008 

i.oi58 

i.o3u8 

1.0457 

£ . 0606 

3 

\    4 

.9245 

.9400 

.9553 

.9706 

.9859 

I • 00 10 

i.oi6i 

£.o3ii 

£.0460 

£.0608 

4 

1  5 

.9248 

.9402 

•  9556 

.9709 

•  9861 

I • 00 1 3 

I. 01 63 

£.03£3 

£.0462 

£.061  £  1 

5 

6 

.9250 

.9405 

.9559 

.971 1 

•  9864 

I ^0015 

1.0166 

£.o3i6 

I.0465 

i.o6i3 

6 

7 

.9253 

.9407 

•  9561 

.9714 

•  9866 

I .0018 

I. 0168 

i.o3i8 

£.0467 

£•0616 

7 

8 
9 

•  9256 

.9410 

.9564 

•9717 

.9869 

I .0020 

I .0171 

I.o32I 

1.0470 

£.0618 

8 

•  9253 

.9413 

•  9566 

•9719 

.987  c 

I ^0023 

1.0173 

£.0323 

£.0472 

£ .o62£ 

9 

lO 

die: 

•  9261 

•94i5 

.9569 

•  9722 

•9874 

I .0025 

I ^0176 

£.0326 

1.0475 

£.0623 

10 
II 

.9263 

.9418 

•  9571 

.9724 

•  9S76 

1.0028 

I •0178 

£.0328 

1-0477 

£.0626 

\   >^ 

•9266 

•  9420 

.9574 

.9727 

.9879 

I .oo3o 

I. 0181 

£.033£ 

£.0480 

£.0628 

12 

i3 

•  9268 

.9423 

.9576 

•  9729 

•  9881 

I .0033 

i.oi83 

£.o333 

£.0482 

£.o63o 

i3 

i4 

.9271 

.9425 

.9579 

•  9732 

.9884 

£•0035 

1.0186 

£.o336 

I -0485 

£.o633 

i4 

i5 

•9274 

.9428 

•  9581 

•  9734 

.9886 

i^oo38 

1.0188 

£.o338 

£.0487 

£-0635 

i5 

1  i6 

.9276 

•9430 

•  9584 

•9737 

•  9889 

I • oo4o 

I .or9i 

i.o34e 

£.0490 

£.o638 

16 

1  "7  i 

.9279 

.9433 

.9587 

.9739 

•  9891 

I ^0043 

I. 0193 

£.0343 

£.0492 

£-0640 

17 

'^1 

•  9281 

•  9436 

.9589 

.9742 

•9894 

i^oo45 

I .0196 

I .o346 

£.0495 

£-0643 

18 

.9284 

•9438 

.9392 

•9744 

•9^97 

I • 0048  j 

1.0198 

i.o348 

£.0497 

1-0645 

19 

1  '^^'■• 

1  21  1' 

.9287 

•9441 

•9J94 

•9747 

.9S99 

I ^0050 

I^020I 

£ .035 I 

£.o5oo 

£-0648 

20 

2£ 

.9289 

•9443 

.9597 

.9750 

•  9902 

i.oo53 

1.0203 

£.0353 

£ .0502 

£.o65o 

?  22  1 

•  9292 

•9446 

.9399 

.9752 

.9904 

I -0055  ! 

I .0206 

£.o356 

£.o5o4 

£.o653 

22 

^  23 

•9294 

•9448 

•  9602 

.9733 

.9907 

1-0058  1 

1.0208 

£.o358 

I .0307 

£.0655 

23 

i  24 

.9297 

•945 1 

•  9604 

•97^7 

.9909 

I ■ 0060 

I ^021  I 

£.036£ 

£.o5o9 

£.o658 

24 

i  25 

.9299 

•9454 

•  9U07 

.9760 

.9912 

1 .0063 

I^02l3 

£.o363 

£.o5l2 

£ • 0660 

25 

a  26 

■  9302 

•  9456 

•  9610 

.9762 

•9914 

I .0063 

I ^0216 

£.o366 

i.o5i4 

£  .  066-2 

26 

27 

•93o5 

•9459 

•9612 

.9765 

.9917 

1 • 0068 

I. 0218 

£.0368 

£.o5£7 

£ . 0663 

27 

28.; 

•  9307 

•  9461 

•9615 

.9767 

•  9919 

1^0070 

I •OQai 

£.0370 

£ .0519 

£ . 0667 

28 

29  \ 

•9310 

•9464 

•9^17 

•977" 

•  9922 

1.0073 

1^0223 

£.0373 

I .o522 

£•0670 

29 

1  ^'^  1 

\  3i  |i 

•  93 1 2 

•  9466 

•  9620 

•9772 

•9924 

£•0075 

1-0226 

I .0373 

£.0524 

£.0672 

3o 
3i 

.9315 

.9469 

.9622 

•9775 

.9927 

I • 0078 

1.0228 

£.0378 

£.0527 

£.0675 

32  1' 

.9317 

.9472 

.9625 

•9778 

•  9929 

I •ooSo 

I.023l 

£.o38o 

£ .0529 

£.0677 

32 

33 

•  9320 

•9474 

.9627 

.9780 

•  9932 

i.oo83 

I -0233 

£.0383 

£.0532 

£.0680 

33 

M 

.9323 

•9477 

.9630 

•  9783 

•99^4 

I • 0086 

1^0236 

£.o3S5 

£.0534 

£.0682 

34 

35 

.9325 

•9479 

-9633 

•9785 

•  9937 

1.0088 

I.0238 

£.o388 

£.0537 

£.0685 

35 

36  J 

.9328 

.9482 

.9635 

.9788 

.9939 

I • 009 [ 

I '0241 

I .0390 

1.0539 

1.0687 

36 

37  1 

.9330 

•9484 

•  9638 

.9790 

•9942 

I .0093 

£•0243 

1.0393 

1.0542 

£.0690 

37 

38 

.9333 

.9487 

•9640 

•  9793 

•9943 

I . Q096 

£•0246 

£ .0395 

I.0544 

£ .0692 

38 

39  1 

■9335 

.9489 

.9643 

•  9793 

•9947 

I . 0098  1 

1.0248 

£.0398 

£.0547 

£ • 0694  j 

39 

'^'^1 

.9338 

.9492 

•9645 

•  9798 

•  9930 

1 .0101  1 

j 

I -0251 

£ • o4oo 

£.0549 

£.0697 

40 
4i 

•9341 

•9495 

.9648 

•  9800 

.9952 

I . 0 I o3 

I .0253 

£.o4o3 

£.o55i 

£.0699 

M^ 

.9343 

•9497 

.9650 

.9803 

•  9933 

I .0106 

1.0256 

i.o4o5 

I.0554 

£ .0702 

42 

43  li 

.9346 

•  9500 

.9653 

•  9805 

•  9937 

I • 0 1 08 

£.0258 

£.o4o8 

£.o556 

£.0704 

43 

\  ^4- 

.9348 

•  9D02 

•  9655 

.9808 

•  9960 

I^OIII ' 

I .0261 

I .04£0 

1.0559 

£•0707 

44 

45 

.9351 

•  95o5 

.9658 

•  0810 

•  9962 

I .01 i3  ' 

£.0263 

£.o4i3 

£.o56i 

1.0709 

45 

46  1 

.9353 

•  9507 

•  9661 

•98i3| 

.996b 

I. 0116 

£.0266 

i.o4i5 

£.0564 

1.07  I  2 

46 

47 

.9356 

•  9510 

•  91363 

•  98 1 6 1 

.9967 

1.0118 

£.0268 

£.04£8 

1.0566 

£.0714  ; 

47 

48 

.9359 

•  951a 

.9666 

•  9818 

•  9970 

I.0I2I 

£.0271 

£ .0420 

£.0569 

1-0717  1 

48 

49 

.9361 

•  951 5 

.9668 

.9821 

.9972 

I ^0123 

£ .0273 

1.0423 

£.0371 

£•0719 

49 

5o 

5i  i 

•9364 
.9366 

.9518 

.9671 

•  9823 

.9973 
•9977 

1 .0126  1 

£.0276 

1.0425 

£.0574 

£ '0721  , 

5o 

5£ 

•  9520 

•9Q73 

.9826 

! 

1.0128 

1.0278 

£.0428 

£.0576 

£•0724  , 

.9369 

.9523 

.9(376 

.9828I 

.9980 

I .oi3i 

£.0281 

£.o43o 

£.0579 

£.0726 

52 

53 

.9371 

.9525; 

•9^78 

•  983i 

.99S2 

I .01 33 

£.0283 

£.0433 

£.058£ 

£.0729 

53 

54  i 

.9374 

.95281 

•  908, 

•  9833 

•  9983 

I -0136 

1.0286 

£.0435 

£.o584 

£.0731  , 

54 

55 

.9377 

.9530' 

•  9083 

•9836 

•  9987 

i.oi38 

1.0288 

£.o438 

£.o586 

1.0734  1 

55 

56 

.9379 

•9533' 

.9686 

•9838 

.9990 

I •oi4i 

I .0291 

i.o44o 

£.0589 

£•0736 

56 

P7 

.9382 

•9536 

•96S9' 

•  9841 

•  9992 

I .0143 

£ -0293 

£.0443 

£.0391 

£.0739 

57 

58  ] 

•  9384 

•  9538 

•9091 

.9843 

.9995 

] .0146 

I .0296 

1.0445 

1.0593 

£.074£ 

58 

1  ^^'■1 

.93S7 

•  9541 

.9O94, 

•  9846 

•9998 

i^oi48 

£.0298 

£.0447 

1.0596 

£.0744 

59 

!  60  [ 

.9389 

.9543,. 9696, 

•  9848 

1 0000 

I  •oiSi 

I  •  c>3o  I 

i.o45o 

£.0598 

£.0746 

60 

i;i 


TABLE    OF    CHORDS:     [I^ai 

7ius  =  1.0U0u]. 

H. 

0 

o' 

65° 

66° 

67° 

68° 

69° 

70° 

71° 

72° 

73° 

1-0746 

1.0893 

I. 1039 

1.1184 

1.1328 

I -1472 

i.i6i4 

I -1756 

1.1896 

I 

1.0748 

1.0895 

1.104 I 

1. 1 1 86 

i.i33i 

1-1474 

I. 1616 

1-1758 

1.1899 

I 

2 

1. 076 1 

1.0898 

1.1044 

I. 1189 

I.I333 

1.1476 

I. 1619 

1  - 1 760 

1 . 1 90 1 

2 

3 

1.0753 

1 . 0900 

I. 1046 

1 . 1 191 

I.I335 

I. 1479 

1.1621 

1-1763 

1.1903 

3 

4 

1.0756 

1.0903 

1.1048 

1 . 1 1 94 

I-I338 

1.1481 

1.1624 

1  - 1 765 

I . 1 906 

4 

5 

1.0758 

I . 0905 

i.io5i 

1 . II 96 

i.i34o 

1. 1 483 

I . 1626 

1-1767 

I . 1 908 

5 

6 

I. 0761 

[•0907 

1-.I053 

I   1198 

1.1342 

I    i486 

1.1628 

I  - 1 770 

1 . 1910 

6 

7 

1.0763 

1.0910 

1.I056 

I  • 1201 

i.iM5 

1.1488 

i.i63i 

I -1772 

1.1913 

7 

8 

I . 0766 

1 . 09 1 2 

I.I058 

1 . 1203 

I-I347 

1.1491 

I.I633 

1-1775 

1-1915 

8 

Q 

1.0768 

1.0915 

I. 1061 

I .1206 

i.i35o 

I. 1493 

1.1635 

I -1777 

1.1917 

9 

lO 

II 

I. 0771 

1.0917 

1.1063 

I. 1208 

I. i352 

I. 1495 

1.1638 

1-1779 

1.1920 

10 
11 

1.0773 

r.0920 

1.I065 

1-1210 

i-i354 

I. 1498 

1 . I 64o 

1-1782 

1.1922 

12 

1.0775 

1.0922 

1.1068 

I.I2l3 

1-1357 

I . i5oo 

1-1642 

I -1784 

I. 1924 

J .. 

i3 

1.0778 

1.0924 

1. 1070 

I.I2l5 

1.1359 

I . l502 

I -1645 

1-1786 

1.1927 

i3 

i4 

1.0780 

1.0927 

1 . 1073 

1. 1218 

i.i362 

i.i5o5 

I. 1647 

1-1789 

1.1929 

i4 

i5 

1.0783 

1.0929 

1.107b 

I . 1220 

1-1364 

i-i5o7 

1. i65o 

1.1791 

1. 193 1 

lb 

i6 

1.0785 

1.0932 

1.1078 

I .1222 

I.I366 

i-i5io 

1.1652 

1-1793 

1.1934 

16 

17 

1.0788 

1.0934 

1.1080 

I. 1225 

1.1369 

I-l5l2 

1.1654 

1-1790 

1-1936 

17 

i8 

1-0790 

1.0937 

1.1082 

I. 1227 

!• 137I 

i-i5i4 

1.1657 

1.1798 

1-1938 

18 

«9 

20 
21 

1-0793 

1.0939 

I.1085 

I.I23o 

I.I374 

i-i5i7 

I. 1659 

I -1800 

1-1941 

19 

1.0795 

1.0942 

1.1087 

I. I  232 

I. 1376 

i-i5i9 

1.1661 

i-i8o3 

1-1943 

20 
21 

1.0797 

1.0944 

I . 1090 

1-1234 

I. 1378 

I  - l522 

1-1664 

i-i8o5 

1-1946 

22 

1 .0800 

1 . 0946 

I .1092 

I . 1237 

i.i38i 

I-I524 

1-1666 

1.1807 

1.1948 

22 

23 

1.0802 

1.0949 

1.1094 

I. 1239 

I.I383 

I-I526 

1-1668 

1.1810 

1.1950 

23 

24 

i.o8o5 

I . 095 1 

1.1097 

I . 1242 

i.i386 

1 . 1529 

I  - 1 67 1 

1.1812 

1.1952 

24 

25 

1.0807 

1.0954 

1.1099 

I. 1244 

1.1388 

i.i53i 

1-1673 

1.1814 

I. 1955 

25 

26 

1 .  08 1 0 

1.0956 

1 .1102 

I. 1246 

I. 1390 

I-I533 

1-1676 

I. 1817 

1.1937 

26 

27 

I. 0812 

1.0959 

1.1104 

1. 1 249 

1.1393 

1-1536 

1-1678 

1.1819 

1.1959 

27 

j8 

i.o8i5 

I. 0961 

1. 1 107 

I.125l 

I. 1395 

1-1538 

1-1680 

1.1821 

1.1962 

28 

20 

I. 0817 

1.0963 

1 . 1 109 

1. 1 254 

I . 1398 

i-i54i 

1-1683 

1.1824 

1.1964 

29 

3o 
3 1 

1.0820 

I • 0966 

1 . 1  III 

I. 1256 

I • i4oo 

I-I543 

1-1685 

1.1826 

1 . 1 966 

3o 
3i 

1-0822 

1.0968 

1.1114 

1. 1 258 

I . l402 

1-1545 

1.1687 

1. 1829 

I. 1969 

32 

1.0824 

I. 0971 

1. 1116 

1-1261 

i.i4o5 

I-I548 

1 . 1 690 

i-i83i 

1.1971 

32 

33 

1.0827 

1.0973 

1.1119 

1.1263 

I. 1407 

i-i55o 

'i .  1692 

1-1833 

1.1973 

33 

34 

1.0829 

1.0976 

I . 1121 

I • 1 266 

1.1409 

1-1552 

I. 1694 

1.1836 

1.1976 

34 

35 

1.0832 

1.0978 

I.II23 

I  - 1 268 

I.l4l2 

I-I555 

1.1697 

1-1838 

1.1978 

35 

36 

I.0834 

1.0980 

I . 1126 

1-1271 

i.i4i4 

1-1557 

I -1699 

1.1840 

I. 1980 

36 

37 

1.0837 

1.0983 

1.II28 

I- 1273 

1-1417 

i.i56o 

1.1702 

1.1843 

1.1983 

37 

38 

1.0839 

1.0985 

i.ii3i 

1-1273 

1-1419 

i.i562 

I  - 1 704 

1.1845 

1.1985 

38 

39 

i.o84i 

1.0988 

I.1I33 

1. 1 278 

I- 1421 

I-I564 

1-1706 

I. 1847 

1.1987 

39 

40 

4i 

1.0844 

1 .0990 

I.II36 

1. 1 280 

1-1424 

1-1567 

1.1709 

i.i85o 

1.1990 

4o 
4i 

1.0846 

1.0993 

1.1138 

I. 1283 

I- 1426 

1-1569 

1.1711 

1-1852 

1.1992 

42 

1.0849 

1.0995 

1 .1140 

1. 1 285 

I -1429 

1-1571 

1.1713 

1-1854 

1-1994 

42 

43 

i.o85i 

1.0997 

£.1143 

1. 1 287 

i-ii3i 

I -1574 

I . 1716 

1-1857 

1-1997 

43 

44 

I.0854 

1. 1000 

I. 1145 

1. 1 290 

1-1433 

I. 1576 

1.1718 

1-1859 

1-1999 

44 

4'^ 

I.0856 

I • 1002 

I. 1148 

1-1292 

I-I436 

I -1579 

1.1720 

I -1861 

I -2001 

45 

4fi 

1.0859 

i.ioo5 

i.ii5o 

I .1295 

1-1438 

i.i58i 

1.1723 

1.1864 

I  - 2004 

46 

47 

I. 0861 

I. I 007 

I.1I52 

1.1297 

i-i44i 

i.i5S3 

1 .1725 

I. 1866 

I  -  2006 

47 

48 

1.0863 

I . lOIO 

1.II55 

1. 1 299 

1-1443 

1.1586 

1-1727 

1.1868 

I • 2008 

48 

49 

1.0866 

1. 1012 

1.1157 

1 .i3o2 

1.1445 

i.i588 

1.1730 

1.1871 

1.2011 

49 

5o 
St 

1.0868 
I. 0871 

1. 1014 

1. 1 1 60 

i.i3o4 

I. I 448 
i.i45o 

1 . 1 590 

1.1732 

1.1873 

1.20 I 3 

5o 
5i 

1.1017 

1.1162 

i.i3o7 

I. 1593 

1.1735 

I -1875 

1 .20l5 

52 

1.0873 

1.1019 

I.II65 

1 . 1 309 

1 . i452 

i.ib9b 

1.1737 

1.1878 

1.2018 

52 

53 

1.0876 

I . 1022 

1.1167 

1  •  i3i I 

1.1455 

1. 1 598 

I. 1739 

1.1880 

1.2020 

53 

54 

1.0878 

1.1024 

I .1169 

i.i3i4 

I. 1457 

1 . 1 000 

I. 1742 

1.1882 

I .2022 

54 

55 

1.0881 

1. 1027 

1.1172 

i.i3i6 

I . 1460 

I. 1602 

I -1744 

1-1885 

1.2025 

55 

56 

1.0883 

1 . 1029 

I.II74 

1 .i3i9 

1.1462 

i.i6u5 

I. 1746 

1.1887 

1-2027 

56 

57 

1.0885 

i.io3i 

1.1177 

I • l321 

I. 1464 

1 . 1 607 

1.1749 

1.1889 

1 -2029 

57 

58 

1.0888 

i.io34 

I. 1179 

i.i323 

I -.4'^"' 

1.1609 

t .1751 

1.1892 

l-2o32 

58 

59 

I .0890 

i.io36 

1.1181 

1.1326 

1.(469 

1..612 

I. 1753 

1.1894 

1-2034 

59 

1  60 

1.089'j 

I .1039 

1.1184 

i.i328 

1-1472 

I .1614 

1.1756 

1.189O 

1 • 2o36 

bo 

14 


TABLE  OF  CHORDS 

:  [Radius  :=1. 

0000]. 

H. 

o' 

74° 

75° 

76° 

77° 

78° 

79° 

§0° 

81° 

82° 

M. 
0' 

i.2o36 

1. 2175 

i.23i3 

1-2450 

1-2586 

1  -2722 

1-2856 

1.2989 

I-3I21 

I 

I • 2039 

1 .2178 

i.23i6 

1.2453 

1-2589 

1-2724 

1-2858 

1.2991 

1-3123 

I 

2 

I .204  I 

1. 2 1 80 

i-23i8 

1.2455 

1-2591 

1 -2726 

1-2860 

1.2993 

1. 3126 

2 

3 

I .2043 

I. 2182 

I -2320 

1-2457 

1-2593 

1-2728 

1-2862 

1 . 2996 

1. 3128 

3 

4 

I . 2046 

1.2184 

1-2322 

1.2459 

1-2595 

1-2731 

1-2865 

1.2998 

I • 3 1 3o 

4 

5 

I . 2048 

I. 2187 

1-2325 

1.2462 

1-2598 

1-2733 

I  -  2867 

1 • 3ooo 

i.3i32 

5 

6 

I • 2o5o 

1. 2 1 89 

1-2327 

I ■ 2464 

1  -  2600 

1.2735 

I  -  2869 

I -3002 

i.3i34 

6 

7 

I -2053 

I .2191 

1-2329 

1  -  2466 

1 -2602 

1.2737 

I. 2871 

1 . 3oo4 

i.3i37 

7 

8 

I .2055 

1 . 2 1 94 

1-2332 

1-2468 

I • 2604 

1  -  2740 

1.2874 

1 . 3007 

i.3i39 

8 

9 

I .2057 

1.2196 

1-2334 

1-2471 

I . 2607 

1-2742 

1.2876 

1 . 3009 

i.3i4i 

9 

lO 

II 

I • 2060 

1 . 2 1 98 

1-2336 

1-2473 

I . 2609 

1-2744 

1.2878 

1.3011 

i.3i43 

10 
11 

I . 2062 

I.220I 

1-2338 

1.2475 

I -2611 

1-2746 

1 . 2880 

i.3oi3 

i.3i45 

12 

I . 2064 

I .2203 

1-2341 

1.2478 

1 .2614 

1.2748 

1.2882 

i.3oi5 

i.3i47 

12 

i3 

I • 2066 

I .2205 

1-2343 

1-2480 

I. 2616 

1.2761 

r.2885 

i.3oi8 

i.3i5o 

i3 

.4 

I -2069 

I .2208 

1-2345 

1-2482 

1.2618 

1.2753 

1.2887 

1.3020 

i.3i52 

i4 

i5 

I. 207 I 

I. 2210 

1.2348 

1-2484 

I .2620 

1.2755 

1.2889 

1.3022 

i.3i54 

i5 

i  '^ 

I . 2073 

I. 2212 

i.235o 

1.2487 

I .2023 

1-2757 

I. 2891 

1.3024 

i.3i56 

16 

17 

1-2076 

I.23l4 

1.2352 

1.2489 

1 .2625 

1 -2760 

1.2894 

1.3027 

i.3i58 

17 

i8 

I • 2078 

I .2217 

1.2354 

I. 2491 

1 .2627 

1 -2762 

1 . 2896 

1.3029 

i.3i6i 

j8 

JQ 

I . 2080 

I .2219 

1.2357 

1.2493 

1 .2629 

1.2764 

I . 2898 

i.3o3i 

i-3i63 

19 

20 
21 

1.2083 

1.2221 

1.2359 

I . 2496 

1-2632 

1.2766 

I • 2900 

i.3o33 

i-3i65 

20 
21 

I.2085 

I .2224 

I.236I 

1.2498 

1.2634 

I • 2769 

1 . 2903 

i.3o35 

1-3167 

22 

1.2087 

1.2226 

1.2364 

1 .25oo 

1.2636 

1.2771 

1 -2905 

i.3o38 

1-3169 

22 

23 

I . 2090 

1.2228 

1.2366 

i.25o3 

1.2638 

1.2773 

1.2907 

1 . 3o4o 

1-3172 

23 

24 

1.2092 

I  .2231 

1.2368 

i.25o5 

I. 2641 

1.2775 

I . 2909 

i-3o42 

I-3I74 

24 

25 

I . 2094 

1-2233 

1-2370 

I .2507 

1.2643 

1.2778 

1.2911 

i-3o44 

1-3.76 

25 

26 

26 

1.2097 

1.2235 

1-2373 

1.2509 

1.2645 

1.2780 

1.2914 

i-3o46 

1-3178 

27 

1.2099 

1.2237 

1-2375 

1.2512 

1 . 2648 

1.2782 

I. 2916 

I • 3o49 

i-3i8o 

27 

28 

1 . 2 1 0  [ 

1.2240 

1.2377 

i.25i4 

1 -2650 

1  -  2784 

1.2918 

i-3o5i 

i.3i83 

28 

29 

I .2104 

1.2242 

1.2380 

i-25i6 

1 .2652 

1-2787 

I -2920 

i-3o53 

i.3i85 

29 

3oj 

I • 2 1 06 

1-2244 

1.2382 

i-25i8 

1-2654 

1.2789 

1 .2922 

i-3o55 

1.3187 

3o 
3i 

3i  ' 

I. 2108 

1.2247 

1.2384 

1 -2521 

1-2656 

1.2791 

1 -2925 

i.3o57 

1.3189 

32 

1 .2111 

1 .2249 

1.2386 

1-2523 

I -2659 

1 .2793 

1.2927 

1  •  3o6;) 

1.3191 

32 

33 

I  .21  l3 

I-225l 

1.2389 

1.2525 

1.2661 

1.2795 

1.2929 

I .3062 

1.3193  i 

33 

34 

I. 21 [5 

1-2254 

I .2391 

1.2528 

1.2663 

1-2798 

1-29JI 

I • 3o64 

1.3196 

34 

35! 

I. 2117 

1 -2256 

1.2393 

1.2530 

I  -  2665 

I • 2800 

1-2934 

1.3066 

I. 3198 

35 

36, 

I .2120 

1-2258 

1-2396 

1-2532 

1-2668 

1-2802 

1-2936 

1.3068 

1.3200 

36 

37' 

I .2122 

1 -2260 

1-2398 

1-2534 

1 -2670 

I . 2804 

1-2938 

1.3071 

1 .3202 

37 

38 

1 .2124 

1-2263 

I -2400 

1-2537 

1 -2672 

I . 2807 

1-2940 

1-3073 

I .3204 

38 

39 

I. 2127 

1-2265 

1 .2402 

1.2539 

1-2674 

1 .2809 

1-2942 

1.3075 

I .3207  ' 

39 

40 

4^! 

I .2129 

1-2267 

i.24o5 

1.2541 

1-2677 

1.2811 

I • 2945 

1.3077 

1.3209 

4o 
41 

I.2l3l 

1-2270 

I . 2407 

1.2543 

1-2679 

1.2813 

I . 2947 

1.3079 

1-321 1 

42 

1.2  I  34 

I  2272 

1.2409 

1.2546 

1-2681 

1.2816 

1  -  2949 

1.3082 

1-3213; 

42 

43 

I. 2  I 36 

1-2274 

I .2412 

1.2548 

1-2683 

1.2818 

I -295 1 

i.3o84 

I-32I5 

43 

44 

I.2I38 

1.2277 

1.2414 

i.255o 

1  -  2686 

1.2820 

I  -  2954 

i.3o86 

1.3218  : 

44 

45 

I .2141 

I .2279 

I .2416 

1-2552 

1-26S8 

1 -2822 

1-2956 

i-3o88 

1-3220 

45 

46 

i.2i43 

1.2281 

I. 2418 

1-2555 

I • 2690 

1-2S25 

I  -  2958 

1 • 3090 

1 -3222 

46 

47 

I .2145 

1.2283 

1.2421 

1-2557 

1-2692 

1-2827 

1  -  2960 

1.3093 

1-3224 

4- 

48 

I.2I48 

1.2286 

1.2423 

1-2559 

1-2695 

1-2829 

1-2962 

1.3095 

1-3226  ' 

48 

49 
So 

5i 

1 . 2 1 5o 

1.2288 

1.2425 

1-2562 

I  -  2697 

1-2831 

1  -  2965 

1.3097 

1-3228 

49 

I.2l52 

1.2290 

1.2428 

1-2564 

I  -  2699 

1-2833 

I  -  2967 

I  -  3099 

1-3231  ! 

5o 
5i 

I. 2  I  54 

1 .2293 

1.2430 

1-2566 

1-2701 

1.2836 

I  -  2969 

i-3ioi 

1-3233 

52 

I.2I57 

1-2295 

1.2432 

1-2568 

1-2704 

1.2838 

1-2971 

i-3io4 

1-3235 

52 

53 

I.  2  I  59 

1  -2297 

1-2434 

1-2571 

I  -  2706 

1.2840 

1-2973 

I .3io6 

1-3237 

53 

54 

I .2161 

1-2299 

1-2437 

1-2573 

1-2708 

1.2842 

1.2976 

1.3108 

i.3>39 

54 

55 

1 . 2 1 64 

1 -23o2 

1-2439 

1-2575 

1-2710 

1.2845 

1.2978 

i.3i 10 

1.3242 

55 

56 

I .2166 

I -23o4 

I -2441 

1-2577 

1-2713 

1.2847 

1 .2980 

1-3112 

1.3244 

56 

57 

1.2168 

i-23o6 

1-2443 

1 -2580 

1 -2715 

1.2849 

1 .2982 

i-3ii5 

1.3246 

57 

58 

I. 2171 

1  -  2309 

1-2446 

1-2582 

1.2717 

1.2851 

1.2985 

i-3ii7 

1.3248 

58 

59 

I. 2  1 73 

I -23 11 

I ; 2448 

1-2584 

1.2719 

1-2854 

1.2987 

1-3119 

1.3250 

59 

60 

1.2175 

i.23i3 

i.245o 

1-2586 

I -2722 

1-2856 

1.2989 

I -3l21 

1.3252  1 

il 

\o 


TABLE 

OF   CHORDS: 

[Radius 

=  1.0000] 

M. 

o' 

83° 

§4° 

§5° 

§6° 

§7° 

88° 

89° 

M. 

0' 

1.3252 

1.3383 

i-35i2 

1.3640 

1.3767 

1.3893 

1.4018 

I 

1.3255 

1-3385 

i.35i4 

1-3642 

1.3769 

1.3895 

1 .4020 

I 

2 

1.3257 

1-3387 

1.3516 

1-3644 

I '3771 

1-3897 

1.4022 

2 

3 

1.3259 

1-3389 

i.35i8 

1.3646 

1.3773 

1.3899 

1 .4024 

3 

4 

1.3261 

1-3391 

1.3520 

1.3648 

1.3776 

1 -3902 

1 .4026 

4 

5 

1-3263 

1-3393 

1-3523 

1.3651 

1.3778 

1-3904 

1 .4029 

5 

6 

1.3265 

1.3396 

1-3525 

1 .3653 

I . 3780 

I -3906 

I • 4o3 I 

6 

7 

1.3268 

1-3398 

1-3527 

1.^655 

1.3782 

1-3908 

1.4033 

7 

8 

1.3270 

1.3400 

1-3529 

i.:;657 

1.3784 

1 -3910 

1.4035 

8 

9 

1.3272 

1.3402 

1-3531 

1.3659 

1.3786 

1-3912 

1.4037 

9  1 

lO 

II 

1.3274 
1.3276 

1.3404 

1-3533 

1-3661 

1.3788 

1-3914 

1.4039 
1.4041 

10  1 

11  ! 

1-3406 

1-3535 

1.3663 

1-3790 

1-3916 

12 

1.3279 

1.3409 

1.3538 

1.3665 

1-3792 

1-3918 

1.4043 

12  1 

i3 

1.3281 

1.3411 

1.3540 

1.3668 

1-3794 

1.3920 

1.4045 

i3 

i4 

1.3283 

i.34i3 

1.3542 

I .3670 

1-3797 

1.3922 

1-4047 

i4 

i5 

1.3285 

i.34i5 

1.3544 

1.3672 

1-3799 

1.3925 

1-4049 

i5 

i6 

1.3287 

i.34r7 

1.3546 

■1.3674 

I-380I 

1.3927 

i-4o5i 

le 

I? 

1.3289 

1.3419 

1.3548 

1.3676 

i.38o3 

1.3929 

i-4o53 

17 

i8 

I .3292 

I. 3421 

1-3550 

1-3678 

i-38o5 

I. 3931 

i-4o55 

18 

'9 

1.3294 

1.3424 

1.3552 

i-368o' 

1-3807 

1.3933 

i.4o58 

19  i 

20 
21 

1.3296 

1-3426 

1.3555 

1-3682 
1-3685 

1-3809 

1-3935 

1 -4o6o 

20     1 
21 

1.3298 

1-3428 

1.3557 

1.3811 

1-3937 

I  -  4062 

22 

x.33(J0 

I  -.34  3o 

1.3559 

I . 3687 

i.38i3 

1-3939 

1-4064 

22 

23 

1.3302 

1.3432 

1.3561 

1-3689 

1.3816 

1-3941 

1-4066 

23 

24 

i.33o5 

1.3434 

1-3563 

I -3691 

i-38i8 

1.3943 

1-4068 

24 

25 

1.3307 

1-3437 

1-3565 

I  -  3693 

I -3820 

1-3945 

1.4070 

25 

26 

1.3309 

1.3439 

1-3567 

1-3695 

1-3822 

1-3947 

1-4072 

26 

27 

I.33II 

I. 3441 

1.3570 

1-3697 

1-3824 

1-3950 

1-4074 

27 

28 

i.33i3 

1-3443 

1-3572 

I • 3699 

1.3826 

1-3952 

1-4076 

28 

29 

i.33i5 

1-3445 

1-3574 

I -3702 

1-3828 

1.3954 

1.4078 

29 

3o 
3i 

1.3318 
I .3320 

1-3447 

1-3576 

1-3704 

1-3830 

1-3956 
1.3958 

i-4oSo 

3o 
3i 

1-3449 

1.3578 

I -3706 

1.3832 

1-4082 

32 

1.3322 

1.3452 

1-3580 

1.3708 

1-3834 

I .3960 

1-4084 

32 

33 

1.3324 

1.3454 

1-3582 

I. 3710 

1-3837 

1-3962 

1-4086 

33 

34 

1.3326 

1.3456 

I. 3585 

I .3712 

1-3839 

1-3964 

1.4089 

34 

35 

1.3328 

1-3458 

1.3587 

1-3714 

I-384I 

1-3966 

I. 4091 

35 

36 

1. 333 1 

1-3460 

1.3589 

1.3716 

1-3843 

1-3968 

1.4093 

36 

3- 

1.3333 

1-3462 

I .3591 

I. 3718 

1.3845 

1-3970 

1.4095 

37 

38 

1.3335 

1-3465 

1-3593 

1 -3721 

1-3847 

1.3972 

1.4097 

33 

39 

1.3337 

1-3467 

1-3595 

1-3723 

1-3849 

1-3973 

1.4099 

39 

40 
4i 

1.3339 

1-3469 

1.3597 

1.3725 
1-3727 

i-3S5i 

1-3977 

i-4toi    • 

40 
4i 

I. 3341 

1-3471 

1.3599 

1-3853 

1-3979 

i-4io3 

42 

1.3344 

1-3473 

I -3602 

1-3729 

1.3855 

1-3981 

i-4io5 

42 

43 

1-3346 

1.3475 

i-36o4 

1.3731 

1-3858 

1-3983 

1-4107 

43 

44 

1.3348 

1-3477 

1  - 36o6 

1-3733 

1-3860 

1-3985 

1-4109 

44 

45 

i-335o 

1.3480 

i-36o8 

1-3735 

1-3862 

1-3987 

I -4iii 

45 

46 

1-3352 

1-3482 

1 -3610 

1.3738 

1-3864 

1-3989 

i-4ii3 

46 

47 

1-3354 

1-3484 

1-3612 

1-3740 

i-386e 

I -3991 

i.4!i5 

4? 

48 

1-3357 

1-3486 

i.36i4 

1.3742 

1-3868 

1-3993 

1-4117 

48 

49 

1-3359 

1-3488 

1.3617 

1-3744 

1.3870 

1-3995 

1.4119 

49 

.  5o 
5i 

I-336I 

1 -3490 

1.3619 

1.3746 

1-3872 

1-3997 

1 -4l22 

5a 
5i 

1-3363 

1-3492 

I. 3621 

1.3748 

1-3874 

1-3999 

I- 4 1 24 

52 

1-3365 

1-3495 

1-3623 

1.3750 

1-3876 

I .4002 

I -4126 

52 

53 

1.3367 

1-3497 

1-3625 

1.3752 

1-3879 

I . 4oo4 

1. 4128 

53 

54 

1.3370 

1-3499 

1-3627 

1.3754 

I.388I 

1.4006 

1 . 4 1 3o 

54 

55 

1.3372 

I-3501 

1-3629 

1.3757 

1-3883 

1-4008 

i.4i32 

55 

56 

1.3374 

i-35o3 

I-363I 

1.3759 

1-3885 

I- 40 10 

1.4134 

56 

57 

1-3376 

I .35o5 

1-3634 

1.3761 

1-3887 

1 . 40 1 2 

i.4i36 

57 

58 

1-3378 

i-35o8 

1.3636 

1.3763 

1  -  3889 

1.4014 

i.4i38 

58 

59 

LiL 

1-3380 

1-3510 

1-3638 

1.3765 

i-389[ 

I .4016 

i-4i4g 

59 

1.3  383 

1-3512 

1-3640 

1-3767 

1 • 3S93 

I. 4018 

1-4142 

60 

Ju 


TABLE  I., 


LOGARITHMS  OF  NUMBERS 


1   TO   10000. 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

I 

0-000000 

26 

1-414973 

5i 

1-707570 

76 

I -880814 

2 

O'3oio3o 

27 

1-431364 

52 

I -716003 

77 

I -886491 

3 

0-477121 

28 

1-447158 

53 

1-724276 

73 

1-892095 

4 

0- 602060 

29 

1-462398 

54 

1-732394 

79 

1-897627 

5 

0-698970 

3o 

i-477'2i 

55 

i-74o363 

80 

1-903090 

6 

0-778151 

3i 

I.-49I362 

56 

I -748188 

81 

1-908485 

7 

0-845098 

32 

I -505150 

57 

1-755875 

82 

i-9i38i4 

8 

0-903090 

33 

i-5i85i4 

58 

1-763428 

83 

1-919078 

9 

0-954243 

34 

I -53 1479 

59 

I -770852 

84 

1-924279 

10 

I -000000 

35 

I  -  544068 

60 

I-778151 

85 

1-929419 

u 

i-o4i3g3 

36 

1-556303 

61 

1-785330 

8b 

1 -934498 

13 

1-079181 

37 

1-568202 

62 

1-792392 

87 

I -939519 
1-944483 

i3 

1 -113943 

38 

1-579784 

63 

1-799341 

88 

14 

1-146128 

39 

I -591065 

64 

I -806180 

89 

1-949390 

i5 

I-176091 

40 

1-602060 

65 

1-812913 

90 

1-954243 

i6 

I -204120 

41 

1-612784 

66 

1-819544 

91 

I -959041 

17 

I  -  230449 
1- 255213 

42 

1-623249 

67 

1-826075 

92 

1-963788 

18 

43 

1-633468 

68 

i-8325og 

93 

1-968483 

»9 

1.278754 

44 

1-643453 

69 

1-838849 

94 

1-973128 

20 

i-3oio3o 

45 

.  I -653213 

70 

1-845098 

95 

1-977724 

21 

1-322219 

1-34242J 

46 

1-662753 

71 

I-85I258 

96 

1-982271 

22 

47 

1-672098 

72 

1-837333 

97 

1-986772 

23 

1-361728 

43 

1-681241 

73 

1-863323 

93 

1-991226 

24 

I-3802II 

49 

1-690196 

74 

1-869232 

99 

1-995635 

25 

1-397940 

5o 

1-698970 

75 

1-875C61 

100 

2-000000 

N.  B.  In  the  following  table,  in  the  last  nine  columns  of  each  page,  -where  the 
first  at  leading  figures  change  from  9'9  to  O's,  the  character  ♦  is  introduced  instead 
of  the  O's,  to  catch  the  eye,  and  to  indicate  that  from  thence  the  annexed  firat 
two  figures  of  tlie  Logaritlim  in  the  second  column  stand  in  the  next  lower  line, 
directly  under  the  asterisk. 


I 

2 

LOGARITHMS  OF  NUMBERS.         Table  I. 

N.  1 

0 

1    2 

S 

4 

5 

6 

7 

8  1  9  1  D.  1 

100 

00  0000 

0434 

0868 

i3oi 

1734 

2166 

2598 

3029 

3461  3891 

432 

131 

4321 

4751 

5iSi 

5609 

6o38 

6466 

6894 

7321 

7748  8174 

428 

132 

»86oo 

9026 

9451 

9876 

♦  3  00 

0724 

1 147 

1570 

1993  24i5 

424 

I03 

01  2837 

3259 

o63o 

4iO0   4321   1 

4940 

536o 

5779 

6197 

6616 

419 

io4 

*7o33 

745 1 

-863 

Sji4  8700 

9116 

9332 

9947 

♦36i 

0775 

416 

io5 

02 1189 

i6o3 

2?:  6 

2428  2841 

3252 

3664  4075 

4486 

4896 

412 

io6 

53o6 

3715  j6i25  ;  6533  j  6942 

7  3  30 

7737 

8j64 

8571 

8978 

408 

iOT 

«9384 

Vstl 

♦  195 

0600  1004 

i4o8 

1812 

2216 

2619 

302I 

404 

!C3 

o3  3424 

4227 

4628  5029 

543o 

5830 

623o 

6629 

7028 

400 

109 

»7426 

7825 

£223 

8620  9017 

9414 

9811 

♦207 

0602 

0993 

396 

no 

04  J  393 

1787 

2182 

2576  2969 

3362 

3755 

4148 

4540 

4932 

393 

111 

5323 

5714 

6io5 

6493 

6883 

7273 

7664 

8o53 

8442 

883o 

389 

:;2 

*92i8 

9606 
3463 

9993 

♦38o 

0766 

11 53 

i533 

1924 

2309 

2694 

336 

ii3 

o5  3078 

3846 

423o 

461 3 

499^ 

5378 

5760 

6142 

6324 

382 

114 

*69o5 

7286 

7656 

8046 

8426 

8803" 

9185 

9563 

9942 

♦  320 

379 

ii5 

06  0698 

1075 

1452 

1829 

2206 

2382 

2958 

3333 

3709 

4o83 

376 

116 

4458 

483n 

5206 

558o 

5953 

6326 

6699 

7071 

7443 

73i5 

372 

i'7 

»8i86 

855; 

8928 

9298 

9668 
3352 

♦o38 

0407 

0776 

1145 

i5i4 

369 

118 

07  1882 

225o 

2617 

2935 

3718 

4o85 

445 1 

4816 

5182 

366 

119 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

*9]8i 

9343 

9904 

♦266 

0626 

0987 

i347 

1707 

2067 

2426 

36o 

12! 

08  2-85 

3i44 

33o3 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

122 

.  ,  6360 

6716 

7071 

7426 

7781 

8i36 

8490 

8843 

9198 

9552 

355 

123 

«99o5 

♦;58 

0611 

0963 

i3i5 

1667 

2018 

2370 

2721 

3071 

35i 

124 

•,93422 

3772 

4122 

4471 

4820 

5169 

55i8 

5866 

6215 

6562 

349 

125 

*69io 

7257 

7604 

7951 

8293. 

8644 

8900 
2434 

9335 

9681 

4026 

346 

126 

100371 

0715 

1059 

i4o3 

«747 

2091 

2777 

3119 

3462 

343 

127 

38o4 

4146 

4487 
7888 

4828 

5169 

55io 

_585i 

6191 

653 1 

6871 

340 

V'H 

»72I0 

-549 

8227 

8563 

8903 

9241 

9579 

9t;i6 

♦253 

338 

129 

1 1  0390 

0926 

1263 

1599 

1934 

2270 

2603 

2940 

3275 

3609 

335 

i3o 

3943 

4277 

461 1 

.4944 

5278 

56n 

5943 

6276 

6608 

6940 

333 

i3i 

.*727i 

7603 

7934 

8265 

8395 

8926 

9256 

9586 

99' 5 

♦245 

33« 

l32 

•  1 2"  0374 

0903 

123l 

i56o 

1888 

2216 

'2544 

2871 

3193 

3525 

32S 

i33 

.3352  ■ 

4178 

45o4 

483o 

oi56 

5481 

.5806 

6i3i 

6456 

6781 

325 

1 34 

.*7io5 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

♦012 

323 

i35 

t3o334 

o655 

0977 

1298 

1619 

1939 

2260 

258o 

2900 

3219 

321 

1 36 

3539 

3358 

4177 

4496 

4814 

5i33 

5451 

3769 

6086 

64o3 

3i8 

13? 

,  -6721  . 

7037 

7354 

7671 

79^7 

83o3 

•8618 

8934 

9249 

9564 

3i5 

i38 

*9379 

♦  191 

o5o3 

0822 

n36 

i45o 

1763 

2076  2389 

2702 

3i4 

189 

i43oi5' 

3327 

3639 

3931 

4263 

4574 

4885 

0196  5507 

58i8 

3n 

J  40 

6128 

6438 

6748 

7o58 

7367 

7676 

7983 

8294  86o3 

Sgii 

309 

141 

*92i9 

9327 

9835 

♦  142 

0449 

0736 

io63 

1370 

1676 

1982 

307 

142 

13  2288 

2594 

2900 

3203 

35 10 

38i5 

4120 

4424 

4728 

5o32 

3o5 

143 

5336 

0640 

5943 

6246 

6549 

6852 

■7154 

7457 

7759 

8061 

3o3 

144 

»8362 

8664 

8965 

9266 

9567 

9868 

♦  168 

0469 

0769 

1068 

3oi 

145 

i6i363 

1667 

1967 

2266 

2564 

^  2863 

3i6i 

3460 

3753 

4o55 

299 

146 

■  ■  4353 

465o 

4947 

5244 

5541 

5838 

6i34 

643o 

6726 

7022 

297 

147 
!48 

-'  7317 

7613 

82o3 

8497 

8792 

9086 

9380 

9674 

9968 

295 

I'J  0262 

o555 

0848 

1141 

1434 

1726 

2019 

23 11 

26o3 

2895 

293 

149 

3iS6 

3478 

3769 

4060 

43  5 1 

4641 

4932 

5222 

55i2 

58o2 

291 

i5q 

6091 

638 1 

6670 

6939 

7248 

7536 

7S25 

8ii3 

84UI 

8689 

289 

i5i 

*8977- 

9264 

9552 

9839 

♦126 

b4i3 

9^9 

0985 

1272' 

1538 

-87 

102 

18  1844 

2129 

24i5 

270c 

2983 
5825 

3270 

3333 

3839 

4123 

4407 

285 

i53 

4691 

4975 

5239 

5542 

6108 

6391 

6674 

6956 

7239 

283 

i54 

*  7321 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

,9771 

♦031 

281 

1 55 

190332 

0612  '  0S92 

1171 

I45i 

1730 

2010 

2289 

2567 
5346 

2846 

279 

1 56 

3125 

34o3  1  368i 

3939 

4237 

45i4 

4792 

5069 

5623 

278 

1 57 

5900 

6176  ,  6453 

6729 

7oo5 

7281 

7336 

7832 

8107 

8382 

276 

n8 

*8657 

8932  9206 

94S1 

9755 

♦029 

o3o3 

0577 

o85o 

1 1 24 

374 

139 

20  i397 

1670  1943 

2216 j  2488 

2761 

3o33 

33o5 

3577 

3843 

272 

U 

' 

1    2 

3  1  4 

5   .  6  1  7 

S  1  9    D.  1 

Tadi.e  I. 

LOGARITHMS  OF  NUMBERS.             3 

i\.  ! 

0 

1 

2 

3 

4   1 

5   j  6 

7 

8 

9 

D. 

i6o 

204120 

4391 

4663 

4934 

5204  1 

5473 

5746 

60.6 

6286 

6556 

271 

i6i 

6826 

7090 

7365 

7634 

7904  1 

8173 

8441 

8710 

8979 

9247 

2(J9 

162 

»95i5 

9783 

♦o5i 

o3i9 

o586 

o853 

1121 

i388 

i654 

1921 

267 

I6i 

21  2188 

2434 

2720 

2986 

3252 

35.3 

3783 

4049 

43.4 

4379 

266 

164 

4844 

5i09 

5373 

5633 

3902 

6166 

643o 

6694 

6957 

7221 

264 

i65 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

166 

22  0108 

0370 

o63i 

0892 

1153 

1414 

1675 

1936 

2196 

2436 

26. 

.67 

271& 

2976 

3236 

3496 

3755 

401 5 

4274 

4333 

4792 

5o5i 

239 

168 

5309 

5568 

5826 

6084 

6342 

6600 

6858 

71.5 

7372 

763o 

238 

169 

*7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

♦193 

256 

i-o 

23  0449 

0704 

0960 

I2I3 

1470 

1724 

•979 

2234 

2488 

2742 

254 

IT 

2996 

3230 

35o4 

3737 

4011 

4264 

4517 

4770 

5o23 

5276 

253 

172 

5528 

5781 

6o33 

6285 

6537 

6789 

7041 

7292 

7544 

7793 

252 

173 

*8o46 

8297 

8548 

8799 

9049 

9:99 

9550 

9800 

♦o5o 

o3oo 

25o 

"74 

24  0349 

0799 

1048 

1297 

1346 

1793 

2044 

2293 

2541 

2790 

249 

175 

3o38 

3286 

3534 

3782 

4o3o 

4277 

4325 

4772 

5019 

5266 

248 

176 

55i3 

5739 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

*7973 

8219 

8464 

8709 

8o-)4 
.393 

9198 
1 638 

9443 

9687 

9932 

♦  176 

245 

1-8 

200420 

0664 

0908 
3J38 

II3I 

1881 

2125 

2368 

2610 

243 

179 

2853 

3096 

3380 

3822 

4064 

43o6 

4548 

4790 

5o3i 

242 

180 

5273 

55i4 

5755 

5996 

6237 

6477 

6718 

69  53 
9355 

7198 

7439 

241 

181 

7679 

7018 

81 58 

8398 

8637 

8877 

91 16 

9594 

9S33 

239 

182 

26  007 1 

o3io 

0548 

0787 

1025 

1263 

i5oi 

1739 

1976 

2214 

238 

i83 

2431 

2688 

2025 

3ie2 

33o9 

3636 

3873 

4109 

4346 

4582 

237 

.84 

4818 

5o54  3290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

i85 

7172 

7406  7641 

7873 

8110 

8344 

3578 

8812 

9046 

9279 

234 

iSi 

»95i3 

9746  9980 

♦  2.3 

0446 

0679 

0912 

1 144 

1 377 

1609 

233 

187 

27  1842 

2074 

23o6 

2538 

2770 

3ooi 

3233 

3464 

3696 

3927 

232 

1S8 

4i58 

4389 

4620 

485o 

5o8i 

53n 

5542 

5772 

6002 

6232  1  23o  1 

189 

6452 

6692 

6921 

7131 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

.  »8754 

8982 

02  u 

943g 

9667 

9895 

♦  123 

o35i 

0578 

0806 

228 

,i<j< 

28  io33 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192 

33oi 

3527 

3753 

3979 

42o5 

443i 

4656 

4882 

5107 

5332 

226 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6go5 

7i3o 

7354 

7578 

223 

194 

7802 

8026 

8249 

8473 

S696 

8920 

9143 

9366 

9589 

9812 

223 

190 

29  oo35 

0257 

0480 

0702 

0923 

1 147 

1369 

1 591 

i8i3 

2o34 

222 

196 

2256 

2478 

2699 

2920 

3i4i 

3363 

3584 

38o4 

4023 

4246 

221 

'97 

4466 

4687 

4907 

3127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198 

6665 

6884 

7104 

7323 

7342 

7761 

7979 

8198 

8416 

8635 

2.9 

199 

*S853 

9071 

92S9 

9307 

9725 

9943 

♦  161 

0378 

0595 

o8i3 

2.8 

200 

3o  io3o 

1247 

1464 

i68i 

1898 

2114 

233 1 

2547 

2764 

2980 

217 

201 

3196 

3412 

3623 

3844 

4039 

4275 

4491 

4706 

4921 

5.36 

2.6 

20  J 

53oi 

5566 

5781 

5996 

6211 

6425 

6639 

877S 

6854 

7068 

7282 

2.5 

2o3 

7496 
*963o 

7710 

7924 

8.37 

835i 

8564 

8991 

9204 

9417 

2.3 

204 

9843 

♦o56 

0268 

0481 

0693 

0906 

lub 

i33o 

i542 

212 

200 

'311734 

1966 

2177 

2389 

2600 

2812 

3o23 

3234 

3445 

3656 

211 

206 

.  3867 

4078 

4289 

4499 

4710 

4920 

5i3o 

5340 

555i 

5760 

210 

207 

5970 

6iSo 

6390 

86^9 

6809 

7018 

7227 

7436 

7646 

7854 

209 

20S 

8o63 

8272 

8481 

889S 

9106 

9314 

9322 

9730 

9938 

208 

209 

32  0146 

o354 

o562 

0769 

0977 

u84 

1391 

1598 

i8o5 

2012 

207 

210 

2219 

2426 

2633 

2839 

3o46 

3252 

3458 

3665 

3871 

4077 

206 

21 1 

4282 

4488 

4694 

4899 
6950 

5io5 

53 10 

55i6 

5721 

5926 

6i3i 

205 

212 

6336 

6541 

6745 

7.55 

7350  ■ 

7563 

7767 

7972 

8176 

204 

2l3 

;  *838o 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

♦oo3 

02 1 1 

2o3 

2  1-i 

,  330414 

0617 

0819 

1022 

1225 

1427 

i63o 

i832 

2o34 

2236 

202 

2l5 

2438 

2640 

2842 

3o44 

3246 

3447 

3649 

385o 

4o5i 

4253 

202 

216 

4454 

4655 

4856 

5o57 

5257 

5458 

5658 

5359 

6o59 

6260 

201 

217 

6460 

6660 

6860 

7060 

7260 

7439 

7659 

7853  8o5S 

8257 

200 

218 

*8456 

8656 

8855 

9054 

9253 

945 1 

9650 

9849.  ♦047 
i83o  2028 

0246 

;? 

219 

340444 

0642 

0841 

io3g 

1237 

1435 

i632 

2225 

X. 

'   0 

1  1 

2 

3 

1  4 

5 

6 

7 

1  ^ 

9 

D. 

4 

LOGARITHMS  OF  NUMBERS.         Table  I. 

N. 

0 

1 

2 

3    4 

6 

6 

7 

8 

9 

D. 

230 

342423 

3620 

2817 
4735 

3oi4 

3212 

3409 

36o6 

38o3 

3999 

4196 

>97 

321 

43q2 

4539 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

196 

222 

6353 

6549 

6744 
8654 

6g39 

7i35 

7330 

7525 

7720 

7915 

8110 

195 

223 

*83o5 

85oo 

8889 

9083 

9-3 

9472 

9666 

9860 

♦o54 

194 

324 

35  0243 

0442 

o!>j6 

0829 

1023 

1216 

i4io 

i6o3 

1796 

1989 

193 

225 

2i83 

2375 

2563 

2761 

2954 
4876 

3i47 

3339 

3532 

3724 

3916 

193 

226 

410S 

43oi 

4493 

4685 

5o68 

5260 

5432 

5643 

5834 

192 

228 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

7?35 
#9833 

8l23 

83i6 

85o6 

8696 

8886 

9076 

9266 

9456 

9646 

190 
189 

329 

♦  025 

02l5 

0404 

0593 

0783 

0972 

1161 

i35o 

1539 

23o 

36  1728 

I9I7 

2io5 

2294 

2482 

2671 

2859 

3o48 

3236 

3424 

188 

23l 

36i2 

38oo 

3988 

4176 

4363 

455 1 

4739 

4926 

5ii3 

53oi 

188 

232 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

J  87 

233 

7356 

7542 

7729 

7913 

8101 

8287 

8473 

8639 
o5i3 

8845 

90  3o 

J  86 

234 

♦  9216 

9401 

9587 

9772 

9958 

♦  143 

o328 

0698 

o883 

i85 

235 

371068 

1253 

1437 

1622 

1806 

1991 

383 1 

2175 

236o 

2544 

2728 

184 

236 

2912 

3096 

3280 

3464 

3647 

401 5 

4193 

4382 

4565 

184 

237 

4748 

4932 

5ii5 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

l83 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8o34 

8216 

j83 

239 

*8398 

858o 

87(11 

8943 

9124 

9306 

9487 

9668 

9849 

♦o3o 

181 

240 

38  021 1 

0392 

0573 

0754 

0934 

iii5 

1296 

1476 

1 656 

1 837 

181 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

242 

38i5 

3903 
3783 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428 

179 

243 

56o6 

5964 

6142 

6321 

6499 

6677 

6856 

7034 
8811 

7212 

178 

244 

7390 

7568 

7746 

7923 

8joi 

8279 

8456 

8634 

8989 

178 

245 

*9i66 

9343 

9520 

9698 

9875 

♦o5i 

0228 

o4o5 

o582 

0759 

177 

246 

39  0935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

176 

247 

2697 

2873 

3o48 

3224 

3400 

3575 

3731 

3926 

4101 

4277 

176 

248 

4432 

4627 

4802 

4977 

5i52 

5326 

55oi 

5676 

585o 

6025 

175 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

25o 

7940 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

95oi 

»73 

25l 

*9674 

9847 

♦020 

0192 

o365 

o538 

071 1 

o883 

io56 

1228 

173 

252 

40  i4oi 

1573 

1745 

I9I7 

2089 

2261 

2433 

26o5 

2777 

2949 
4663 

173 

253 

3l2I 

3292 

3464 

3635 

3807 

3978 

4149 
5858 

4320 

4492 

171 

254 

4834 

5oo5 

5176 

5346 

55i7 

5688 

6029 

6199 

6370 

171 

255 

6540 

6710 

6881 

7o5i 

7221 

7391 
9087 

7561 

773i 

7901 

8070 

170 

-256 

8240 

8410 

8579 

8749 

8918 

9257 

9426 

9395 

9764 

169 

257 

»  9933 

♦  102 

0271 

0440 

0609 
2293 

0777 

0946 

1114 

1283 

i45i 

169 

258 

41  1620 

1788 

1956 

2124 

2461 

2629 
43o5 

2796 

2964 

3i32 

168 

259 

33oo 

3467 

3635 

38o3 

3970 

4137 

4472 

4639 

4806 

167 

260 

4973 

5i4o 

53o7 

5474 

5641 

58o8 

5974 

6141 

63o8 

6474 

167 

261 

6641 

6807 

6973 

V'^ 

7306 

7472 

7638 

7804 

7970 

8i35 

J  66 

262 

83oi 

8467 

8633 

S798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

263 

♦  9956 

♦  121 

0286 

0431 

0616 

0781 

0945 

mo 

1275 

1439 

i65 

264 

42  1604 

1768 

1933 

2097 

2261 

2426 

2390 

2754 

2918 

3o82 

164 

265 

3246 

3410 

3574 

3737 

3901 

4o65 

4228 

4392 

4555 

4718 

164 

266 

4882 

5o45 

5208 

5371 

5534 

5697 

586o 

6023 

6186 

6340 
7973 
9391 

1 63 

267 

65ii 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

781 1 

162 

268 

8i35 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

163 

269 

•  9752 

9914 

♦073 

0236 

0398 

0559 

0720 

0881 

1042 

I203 

161 

270 

43  i364 

i525 

1685 

1846 

3007 

2167 

2328 

3488 

2649 

2809 

161 

271 

2969 

3i3o 

3290 

345o 

36io 

3770 

3930 
5526 

4090 

4249 

4409 

160 

272 

4569 
6i63 

4729 

4888 

5o48 

5207 

5367 

5685 

5844 

6004 

159 

273 

6322 

6481 

6640 

g§l 

6957 

7116 

7275 
8839 

7433 

7592 

i5q 

i58 

574 

7751 

7909 

8067 

8226 

8343 

8701 

9017 

9175 

375 

*9333 

9491 

9643 

qSo6 

9964 

♦  123 

0279 

0437 

0594 

0753 

i58 

276 

440909 

1066 

1224 

i38i 

1338 

1695 

l832 

2009 
3576 

3166 

3323 

157 

277 

3480 

2637 

2793 

2o5o 
43i3 

3io6 

3263 

3419 

3732 

3889 

1 57 

278 

4045 

4201 

4357 

4669 
6226 

4825 

4981 

5i37 

5293 

5440 
7003 

1 56 

379 

56o4 

5760 

5915 

6071 

6382 

6537 

6692 

6843 

i55 

N. 

0 

1 

0 

3 

4 

5 

6 

7 

8 

9 

D. 

Table  I. 

LOGARITHMS  OF  NUMBERS. 

5 

N. 
280 

0 

1 

2 

3 

4 

5 

6 

1 

8 

9 

D. 

44  7'58 

73i3 

8861 

7463 

7623 

7778 

7933 

8088 

8242 

8397 

8532 

1 55 

281 

#8706 

90i5 

9170 

9324 

9478 

5633 

9787 

9941 

♦095 

1 54 

282 

45  0249 

o4o3 

0557 

071. 

o865 

1018 

1172 

i326 

'479 

i633 

1 54 

283 

1786 

1940 

2093 

1247 

2400 

2553 

2706 

2859 

3oi2 

3i65 

1 53 

284 

33i3 

3471 

3624 

3777 

3g3o 

4082 

4235 

4387 

4540 

4692 

i53 

283 

4845 

nu 

5i5o 

53o2 

5-454 

56o6 

5758 

5910 

6062 

6214 

l52 

286 

6366 

6670 

6821 

6973 

7.25 

8638 

7276 

7428 

7579 

773i 

l52 

288 

7882 

8o33 

8184 

8336 

8487 

8789 

8940 

9091 

9242 

i5i 

#9392 

9543 

9694 

9845 

9995 

♦  146 

0296 

0447 

05q7 

0748 

i5i 

289 

46  0898 

1048 

1198 

1 348 

1499 

1649 

1799 

1948 

2098 

2248 

i5o 

290 

2398 

2548 

2697 

2847 

2997 

3i46 

3296 

3445 

3594 

5o85 

3744 

i5o 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

3234 

149 

292 

5383 

5532 

568q, 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

148 

293 

6868 

7016 

7164 

73 12 

7460 

7608 

7756 

7904 
9380 

8032 

8200 

294 

8347 

8495 

8643 

8790 

8938 

9083 

9233 

9527 

9675 

148 

295 

♦  9822 

9969 

♦  116 

0263 

0410 

0557 

0704 

o85i 

0998 

1145 

147 

296 

47  1292 

1438 

1 585 

1732 

1878 

2025 

2171 

23i8 

2464 

2610 

146 

297 

27D6 

2903 

3o49 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

298 

4216 

4362 

45o8 

4633 

4799 

4944 

5090 

5235 

5381 

5526 

146 

299 

5671 

58i6 

5962 

6107 

6232 

6397 

6542 

6687 

6832 

6976 

145 

3oo 

7121 

7266 

7411 

7555 

7700 

7844 

7989 

8i33 

8278 

8422 

145 

3oi 

8566 

8711 

8855 

8999 

9143 

9287 

943 1 

9575 

97'9 

9S63 

144 

3o2 

48  0007 

oi5i 

0294 

0438 

o582 

0725 

0869 

1012 

ii56 

1299 

2731 

144 

3o3 

1443 

1 586 

1729 

1872 

2016 

2159 

2  3  02 

2445 

2588 

143 

3o4 

2874 

3oi6 

3i59 

33o2 

3445 

3587 

3730 

3872 

4oi5 

4157 

143 

3o5 

43oo 

4442 

4585 

4727 

4869 

5oii 

5i53 

5295 

5437 

5579 

142 

3o6 

5721 

5863 

6oo5 

6147 

6289 

643o 

6572 

6714 

6S55 

6997 

142 

3o7 

7i38 

7280 
8692 

7421 

7563 

7704 

7845 

7086 
9396 

8127 

8269 

8410 

141 

3o8 

855i 

8833 

8974 
o38o 

9114 

9255 

9537 

9677 

9818 

i4i 

3o9 

#9958 

♦099 

0239 

0520 

0661 

0801 

0941 

1081 

1222 

140 

3io 

49  i362 

l502 

1642 

1782 

1922 
3i.9 

2062 

2201 

2341 

2481 

2621 

140 

3ii 

2760 

2900 

3o4o 

3179 

3458 

3597 

3737 

3876 

401 5 

139 

3l2 

4 1 55 

4294 

4433 

4572 

47 1 1 

485o 

49«9 

5128 

5267 

5406 

1 39 

3i3 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

65i5 

6653 

6791 

139 
1 38 

3i4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8o35 

81-3 

3i5 

83ii 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

i38 

3i6 

#9687 

9824 

9962 

♦099 

0236 

0374 

o5ii 

0648 

0785 

0922 

i37 

3i7 

5o  1059 

1196 

i333 

1470 

1607 

1744 

1880 

2017 

2i54 

2291 

137 

3i8 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

35i8 

3635 

i36 

319 

3791 

3927 

4o63 

4199 

4335 

4471 

460-j 

4743 

4878 

5oi4 

i36 

320 

5i5o 

5286 

5421 

5557 

5693 

5828 

5964 
7316 

6099 

6234 

6370 

i36 

321 

65o5 

6640 

6776 

6911 

7046 

7181 

745i 

7586 

7721 

i35 

322 

7856 

799  > 
9337 

8126 

8260 

8393 

853o 

8664 

8799 

8934 

9068 

i35 

323 

»92o3 

9471 

9606 

9740 

9874 

4009 

0143 

0277 

0411 

1 34 

324 

5i  o545 

0679 

o8i3 

0947 

1081 

1213 

1 349 

1482 

1616 

1750 

i34 

325 

1 883 

2017 

2l5l 

2284 

2418 

255l 

2684 

2818 

2931 

3084 

i33 

326 

32i3 

335i 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4414 

i33 

327 

4548 

4681 

48i3 

4946 

5079 
64o3 

52  1 1 

5344 

5476 

56og 

5741 

i33 

328 

5874 

6006 

6139 

6271 

6535 

6668 

6800 

6932 

7064 

l32 

329 

7196 

7328 

7460 

7392 

7724 

7855 

7987 

8119 

825i 

8382 

l32 

33o 

85i4 

8646 

8777 

8909 

9040 

9171 

93o3 

9434 

9566 

9697 

i3i 

33i 

*9828 

9959 

♦ogo 

0221 

o353 

0484 

061 5 

0745 

0876 

1007 

i3i 

332 

52II38 

1269 

1400 

i53o 

1 66! 

1792 

1922 

2o53 

2i83 

23i4 

i3i 

333 

2444 

2373 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

36i6 

i3o 

334 

3746 

3876 

4006 

4i36 

4266 

4396 

4526 

4656 

4785 

4915 

i3o 

335 

5o45 

5174 

53o4 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

336 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

-5372 
8660 

7301 

129 

^.ll 

763o 

7759 

7888 

8016 

8145 

8274 

8402 

853 1 

8788 

129 

338 

#8917 

9043 

9'74 

93o2 

9430 

9559 

9687 

9815 

9943 

♦072 

128 

33, 

53  0200 

o328 

0456 

o584 

0712 

0840 

0968 

1096  1  1223 

i35i 

128 

N. 

0 

1 

0 

1  3    4 

5  1  6 

7    8 

9 

D. 

6 

LOGARITHMS  OF  NUMBERS.         Table  I. 

Xj 

0    j 

1    2 

3 

4 

5     6  ! 

7 

8 

9 

D. 

340  ! 

53  1479 

1607 

1734 

1862 

iggo 

2.17 

2245  ' 

2372 

25oo 

2627 

128 

341 

2754 

2882 

3  cog 

3i36 

3264 

3391 

33i8 

3645 

3772 

3899 

»27 

342 

4026 

4i53 

4280 

4407 

4534 

4661 

4787 

4gi4 

5o4i 

5167 

127 

343 

5294 

5421 

5547 

5674 

58oo 

5927 

6o53 

6180 

63o6 

6432 

126 

344 

6558 

6685 

681 1 

6937 

7063 

7189 

73i5 

7441 

7367 

7693 

126 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

869g 

8825 

8951 

126 

346 

*9076 

9202 

g327 

9452 

9578 

9703 

9829 

9934 

♦979 

0204 

123 

347 

540329 

0455 

o58o 

0705 

o83o 

og53 

1080 

I205 

i33o 

1454 

125 

343 

1579 

1704 

i82g 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

349 

2823  . 

2900 

3074 

3igg 

3323 

3447 

3571 

36g6 

3820 

3944 

124 

35o 

4068 

4192 

43i6 

4440 

4564 

4688 

4812 

4g36 

5o6o 

5i83 

124 

35i 

5307 

543 1 

5555 

5678 

58o2 

5g25 

6049 

6172 

6296 

64.9 

124 

352 

6543 

6666 

6789 

6gl3 

7o36 

7139 

7282 

74o5 

7520 
8758 

7632 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

85i2 

8635 

8881 

123 

354 

*goo3 

9126 

9249 

9371 

94g4 

9616 

9739 

9861 

9984 

♦  106 

123 

355 

55  0228 

o35i 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

i328 

122 

356 

l45o 

1572 

i6g4 

I8I6 

1938 

2060 

2181 

23o3 

2423 

2547 

122 

357 

2668 

2790 

2gii 

3o33 

3 1 55 

3276 

3398 

35ig 

3640 

3762 

121 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

473i 

4852 

4973 

121 

359 

5og4 

52i5 

53J6 

5457 

5578 

5699 

5820 

5g4o 

6061 

6182 

121 

36o 

63o3 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

120 

36 1 

7507 
8709 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

362 

8829 

8948 

9068 

9188 

9308 

9428 

g548 

9667 

9787 

120 

363 

*9907 

♦026 

0146 

0265 

o385 

o5o4 

0624 

0743 

o863 

0982 

119 

364 

56  1101 

1221 

1 340 

i45g 

1 5-8 

1698 

1817 

ig36 

2033 

2174 

119 

365 

2293 

2412 

253i 

265o 

2769 

2887 

3  006 

3i25 

3244 

3362 

119 

366 

3481 

36oo 

3718 

3837 

SgSD 

4074 

4192 

43ii 

4429 

4548 

"2 

367 

4666 

4-84 

4903 

5o2i 

5i3g 

3257 

5376 

54g4 

3612 

5730 

118 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6009 

118 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

784g 

7967 

8084 

118 

370 

8202 

83 19 

8436 

8554 

867J 

8788 

8905 

go23 

9140 

9257 

117 

371 

*9374 

94gi 

g6o8 

9725 

g842 

9959 

♦076 

0193 

o3o9 

0426 

117 

372 

57  o543 

0660 

0776 

0893 

lOlO 

1 126 

1243 

1359 

2523 

1476 

1592 

117 

373 

1709 

1825 

1942 

2038 

2174 

22gi 

2407 

2639 

2735 

116 

374 

2872 

2988 

3 104 

3220 

3336 

345.2 

3568 

3684 

38oo 

3915 

116 

375 

4o3i 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

376 

5 1 88 

53o3 

54ig 

5534 

565o 

5763 

588o 

5gg6 

61U 

6226 

ii5 

377 

6341 

645- 

6572 

6687 

6802 

6917 

7o32 

7147 

7262 

7377 

ii5 

378 

7492 

7607 

7722 

7836 

7901 

8066 

8181 

82g5 

8410 

8525 

U5 

379 

86J9 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

1x4 

38o 

*97^4 

9898 

♦012 

0126 

0241 

0355 

0469 

o583 

0697 

081 1 

114 

38i 

58  0925 

io39 

ii53 

1267 

1 38 1 

1495 

1608 

1722 

i836 

1950 

114 

382 

2o63 

2177 

22gl 

2404 

25i8 

263 1 

2745 

2858 

2972 

3o85 

114 

383 

3199 

33i2 

3426 

353g 

3032 

3763 

3879 

3992 

4io5 

4218 

ii3 

384 

433 1 

4444 

4557 

4670 

4783 

4896 

5oog 

5l22 

3235 

5348 

ii3 

385 

5461 

5574 

5686 

5799 

0912 

6024 

6137 

625o 

6362 

6475 

ii3 

386 

6587 

6700 

6812 

6q25 

7037 

7149 

7262 

7374 

7486 

7599 

112 

387 

77'' 

7823 
§944 

7g35 

8047 

8160 

8272  8384 

8496 

8608 

8720 

113 

388 

8832 

90  56 

9167 

9279 

9391  95o3 

9615 

9726 

9838 

113 

389 

«995o 

♦061 

01-3 

0284 

o3g6 

o5o7  0619 

0730 

0842 

0953 

113 

390 

59  io65 

1 176 

1287 

1 399 

1310 

1621  1732 

1843 

1933 

2066 

III 

3gi 

2177 

2288 

2399 

25l0 

2621 

2732  2843 

2934 

3o64 

3175 

111 

392 

3286 

3397 

35o8 

36i8 

372g 

3840  3950 

4061 

4171 

4282 

111 

393 

4393 

45o3 

4614 

4724 

4834 

4945  5o55 

5i65 

5276 

5386 

no 

394 

5496 

56o6 

5717 

0827 

5g37 

6047 

6157 

6267 

6377 

6487 

no 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

no 

396 

7695 

7S05 

7914 

8024 

81 34 

8243 

8353  8462 

8572 

8681 

no 

397 

8791 

8900 

9009 

9"9 

9228 

9337 

g446  g556 

9663 

9774 

109 

398 

»9883 

9992 

♦  101 

0210 

o3ig 

0428 

0337  0646 

0755 

0864 

109 

399 

60  0973 

1082  1  1191 

1299  1408 

i5i7 

1625  1734 

1843 

1951 

109 

N. 

0 

1    2 

3  1  4 

1   5    6  1  7    8 

9 

D. 

Table  I. 

LOGARITHMS  OF  NUMBERS. 

7 

N. 

0 

1    2 

3  1  4 

5 

6    7 

8 

9 

D. 

400 

602060 

2169 
3253 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3o36 

108 

401 

3i44 

336i 

3469 

3577 

3686 

3794 

3902 

4010 

4118 

108 

402 

4226 

4334 

4442 

455o 

4653 

4766 

4874 

4982 

5089 

5)97 

108 

4o3 

53o5 

54i3 

5521 

5628 

5736 

5844 

5951 

6059 
7133 

6166 

6274 

108 

404 

633i 

6489  6396 

6704 

6811 

6919 

7026 

7241 

7348 

107 

4o3 

7455 

7562  7669 

7777 

7884 

799 « 

8098 

8205 

83i2 

8419 

107 

406 

8526 

8633  8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

107 

407 

*95o4 

9701  9808 

9914 

♦021 

0123 

0234 

o34i 

0447 

o554 

107 

408 

61  0660 

0767  1  0873 

0979 

1086 

1152 

1298 

i4o5 

i5ii 

1617 

106 

409 

1723 

1829!  1936 

2042 

2148 

2234 

236o 

2466 

2572 

2678 

106 

410 

2784 

2890  2996 

3 1 02 

3207 

33i3 

3419 

3525 

363o 

3736 

106 

4ii 

3SJ2 

3947  ;  4o53 

4159 

4264 

4370 

4473 

458i 

4686 

4792 

106  . 

412 

4S97 

Coo3  1  5io8 

52i3 

5319 

5424 

5529 

5634 

5740 

5845 

io5 

4i3 

5930 

6o55 

6160 

6265 

6370 

6476 

658i 

6686 

6790 

6895 

io5 

414 

7000 

7105 

7210 

73i5 

7420 

7325 

7629 

7734 

7839 

7943 

io5 

4i5 

8oj8 

8i53 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

io5 

416 

*  9093 

9198 

9302 

9406 

931 1 

9615 

9719 

9824 

9928 

♦o32 

104 

417 

62  oi36 

0240 

o344 

0448 

o552 

o656 

0760 

0864 

0968 

1072 

104 

418 

1 176 

1280 

1 384 

1483 

1592 

1695 

•799 

1903 

2007 

2110 

104 

419 

2214 

23i3 

2421 

2523 

2628 

2732 

2833 

2939 

3o42 

3i46 

104 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3S69 

3973 

4076 

4179 

io3 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5oo4 

5io7 

5210 

io3 

422 

53i2 

541 5 

55i8 

5621 

0724 

5827 

5929 

6o32 

6i35 

6238 

io3 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7o58 

7161 

7263 

io3 

424 

7366 

7463 

757, 

7673 

7775 

7873 

7980 

8082 

8i85 

8287 

102 

425 

83?9 

8491 

8393 

8693 

8797 

8900 

9002 

9104 

9206 

93oS 

102 

426 

*94io 

9512 

9613 

97i5 

9817 

9919 

♦021 

0123 

C224 

o326 

102 

427 

63  o4'8 

o53o 

o63i 

0733 

o835 

0936 

io38 

1139 
2i53 

1241 

i342 

102 

-428 

1444 

1 545 

1647 

1748 

1849 

1931 

2052 

2255 

2336 

101 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3o64 

.3 1 65 

3266 

3367 

101 

43o 

3468 

3569 

3670 

3771' 

3872 

3973 

4074 
5o8i 

4175 

4276 

4376 

100 

43i 

44-'7 

4578 

4679 

4779 

4880- 

4981 

5182 

5283 

5383 

100 

432 

5484 

5584 

5683 

0785 

5886 

5986 

6087 

6181 

6287 

6388 

100 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 
83  »9 

100 

434 

7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8290 

99 

435 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

92S7 

9387 

99 

436 

»948& 

9586 

9686 

9783 

9885 

9984 

♦084 

oi83 

0283 

o382 

99 

437 

640481 

o58i 

0680 

0779 

0879 

0978 

1077 

1177 
2168 

1276 

.1375 

99 

438 

1474 

1573 

1672 

I77« 

1871 

1970 

2069 

2267 

2366 

99 

439 

2463 

2563 

2662 

2761 

2860 

2939 

3o58 

3i56 

3255 

3354 

99 

440 

3453 

355i 

365o 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

98 

441 

4439 

4537 

4636 

4734 

4832 

4931 

5o29 

5i27 

5226 

5324 

98 

442 

5422 

5521 

5619 

57.7 

58i5 

5913 

6011 

6110 

6208 

63o6 

98 

443 

6404 

65o2 

6600 

6698 

6796 

6894 

6992 

7089 

V^l 

7285 

9^ 

444 

7383 

7481 

7379 

7676 

7774 

7872 

7969 

8067 

8i65 

8262 

98 

445 

836o 

8453 

8555 

8653 

G750 

8843 

8945 

9043 

9140 

9237 

97 

446 

*9335 

9432 

9530 

9627 

9724 

9821 

9919 

♦016 

oii3 

0210 

97 

447. 

65  o3o8 

o4o5 

o5o2 

0399 

0696 

0793 

0890 

0987 

1084 

1181 

97 

448 

1278 

1375 

1472 

ID69 

1666 

1762 

1839 

1956 

2o53 

2130 

97 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3ii6 

97 

45o 

32i3 

3309 
4273 

34o5 

3302 

3598 

3695 

3791 

3888 

3984 

4080 

96 

45i 

4177 

4369 

4465 

4562 

4658 

4734 

485o 

4946 

5o42 

96 

432 

5i3S 

5235 

533 1 

5427 

5523 

5619 

5710 

58io 

5906 
6864 

6002 

96 

453 

6oq3 

6194 

6290 

6386 

6482 

6377 

6673 

6769 

6960 

96 

454 

7o56 

7152 

7247 

7343 

7433 

7534 

7629 

7723 

7820 

7916 

96 

455 

801 1 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

9? 

456 

8965 

9060 

9155 

9200  9346 

9441 

9536 

9631 

9726 

9821 

93 

407 

»  9916 

♦on 

0106 

0201  0296 

0391 

0486 

o58i 

0676 

077' 

93 

458 

66  o865 

0960 

io35 

ii5o  1245 

1339 

1434 

l529 

1623 

1718 

95 

459 

i8i3 

1907 

2002 

: 2096  2191 

2286 

238o 

2475 

2569 

2663 

93 

N. 

0 

1 

2  1  3  1  4 

5    6  1  7 

8 

9 

D. 

n 


8 

LOGARITHMS  OF  NUMBERS.         Table  I. 

460 

0 

1 

2 

3 

4 

5 

6 

3324 

7 
3418 

8 

9 

D. 

66  2738 

2852 

2947 

3o4i 

3i35 

323o 

35i2 

3607 
4548 

94 

461 

3701 

3705 
4736 

3889 

3983 

4078 

4172 

4266 

436o 

4454 

94 

462 

46.'i2 

483o 

4924 

5oi8 

5ll2 

5206 

5299 

5393 
633 1 

5487 

94 

463 

558 1 

5675 

5769 
6705 

5862 

5f)56 

6o5o 

6143 

6237 

6424 

94 

464 

65i8 

6612 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

94 

465 

7453 

7546 

7640 

7733 
8665 

7826 

7920 

8oi3 

8106 

8199 

8293 

93 

466 

8386 

8479 

8572 

8759 

8852 

8945 

9o38 

9i3i 

9224 

93 

467 

*93i7 

9410 

95o3 

9596 

9689 

9782 

9S75 

0895 

♦060 

oi53 

93 

468 

67  0246 

o33g 
1265 

043 1 

o524 

0617 

0710 

0802 

0988 

1080 

93 

469 

1173 

i358 

i45i 

1 543 

1636 

1728 

1821 

1913 

2005 

93 

470 

2098 

2190 

2283 

2375 

2467 

256o 

2652 

2744 

2836 

2929 
385o 

92 

471 

3o2i 

3ii3 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

92 

472 

3942 

4o34 

4126 

4218 

43io 

4402 

4494 

4586 

4677 

4769 

92 

473 

486i 

4953 

5o45 

5i37 

5228 

5320 

5412 

55o3 

5595 

5687 

92 

474 

5778 

5870 

5962 

6o53 

6145 

6236 

6328 

6419 

65ii 

6602 

92 

475 

6694 

6785 

6876 

7789 
8700 

6968 

7059 

7.5i 

7242 
8i54 

7333 

7424 

75i6 

91 

476 

7607 

7698 

7881 
8791 

7972 
8882 

8o63 

8245 

8336 

8427 

91 

477 

85i8 

8609 

8073 

9882 

9064 

9155 

9246 

9337 

91 

478 

»9428 

9519 

9610 

9700 

979 « 

9973 

♦o63 

01 54 

0245 

91 

479 

68  o336 

0426 

o5(7 

0607 

0698 

0789 

0879 

0970 

1060 

ii5i 

91 

480 

1241 

i332 

142? 

i5i3 

i6o3 

1693 

1784 

1874 

1964 

2o55 

90 

481 

2145 

2235 

2320 

2416 

25c/6 

2596 

2686 

2777 

2867 

2957 

90 

•482 

3o47 

3i37 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3657 

90 

483 

3o47 

4o37 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

484 

4845 

4935 

5o25 

5ii4 

5204 

5294 

5383 

5473 

5563 

5652 

90 

485 

5742 

583 1 

5921 
68i5 

6010 

6100 

6189 
7083 

6279 

6368 

6458 

6547 

89 

486 

6636 

5726 

6904 

6994 

7172 
8064 

7261 

735i 

7440 
833 1 

89 

487 

7529 

7618 

7707 
85o8 
9486 

8687 

7886 
8776 

Mil 

8i53 

8242 

89 

48S 

8420 

85o9 
939S 

8953 

9042 

9i3i 

9220 

89 

489 

*93o9 

9575 

9664 

9753 

9841 

9930 

♦019 

0107 

89 

490 

690196 

0285 

0373 

0462 

o55o 

0639 

0728 

0816 

0905 

0993 

89 

491 

1081 

1 1 70 

1258 

1 347 

1435 

i524 

1612 

1700 

1789 

1877 

8S 

492 

1965 

■  2847 

2o53 

2142 

223o 

23i8 

2406 

2494 

2583 

2671 

2759 

88 

493 

2935 

3o23 

3iii 

3199 

3287 

3375 

3463 

355i 

3639 

88 

494 

3727 

38i5 

3go3 

3991 

4078 

4166 

4234 

4342 

443o 

4517 

88 

495 

460  5 

46g3 

4781 

4868 

4956 

5o44 

5i3i 

5219 

5307 

5394 

88 

496 

5482 

5569 

5657 

5744 

5832 

6793 

6007 

6094 

6182 

6269 

87 

497 
49^ 

6356 

6444 

653 1 

6618 

6706 

6880 

6968 

7055 

7142 
8014 

87 

7229 

7317 

7404 

7491 

7578 
8449 

7665 

7752 

7S39 
8709 

7926 
8796 

87 

499 

8101 

8188 

8275 

8362 

8535 

8622 

8683 

87 

5oo 

8970 

9057 

9144 

923 1 

9317 

9404 

9491 

9578 

9664 

975 1 

^7 

5oi 

»9838 

9924 

♦oil 

0098 

0184 

0271 

o358 

0444 

o53i 

0617 

U 

502 

70  0704 

0790 

0877 

0963 
1827 

io5o 

ii36 

1222 

1 309 

1395 

2258 

1482 

5o3 

1 568 

1 654 

1741 

1913 

1999 

2086 

2172 

2344 

86 

5o4 

243i 

2317 

26o3 

2689 

2775 

2861 

2947 

3o33 

3119 

32o5 

86 

5o5 

3291 

3377 

3463 

3540 

3635 

3721 

3807 

3895 

3979 

4o65 

86 

5o6 

4l3l 

4236 

4322 

4408 

4494 

4579 
5436 

4665 

4751 

483t 
5693 

4922 

86 

5o7 

5oo8 

5094 

5179 

5265 

5330 

5522 

5607 

5778 

86 

008 

5864 

5949 
68o3 

6o3d 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

509 

6718 

6888 

6974 

7059 

7144 

7229 

73i5 

7400 

7485 

85 

5io 

7570 

7655 

7740 

7826 

79" 

8761 

8846 

8081 

8166 

825i 

8336 

85 

5ii 

8421 

85o6 

8591 

8676 

8931 

9°'5 

9100 

9185 

85 

5l2 

*9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

♦o33 

85 

5i3 

71  0117 

0202 

0287 

0371 

0456 

o54o 

0625 

0710 

0794 
1639 

0870 
1723 

85 

5i4 

0963 

1048 

Il32 

1217 

i3oi 

i385 

1470 

1554 

84 

5i5 

1807 

1892 

1976 

2060 

2144 

2229 

23i3 

SS 

2481 

2566 

84 

5i6 

265o 

2734 

2S18 

2902 

2986 
3826 

8070 

3i54 

3323 

3407 
4246 

84 

5i7 

3491 

3575 

365o 

3742 

3910 

3004 
4833 

4078 

4162 

84 

5i8 

433o 

4414 

iitl 

458i 

4665 

4749 

4916 

5ooo 

5084 

84 

5i9 

5167 

525i 

54-18 

55o2 

5586 

5669 

5753 

5836 

5920 

84 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

Table  I. 

LOGARITHMS  OF  NUMBERS.             »  j 

N. 

0   1 

1    2  1 

3    4 

5 

6 

7 

8 

9 

D. 

520  ! 

71  6oo3 

6087 

6170 

6254  6337 

6421 

65o4 

6588 

6671 

6754 

83 

521 

6833 

6921 

7004 

7088 

7171 

7234 

7338 

7421 

7504 
8336 

7587 

83 

522 

7671 

7754 

8668 

ir^i 

8oo3 

8086 

8169 

8253 

8419 
9248 

83 

523 

85o2 

8585 

8834 

8917 

9000 

9083 

9165 

83 

524 

»933i 

9414 

9497 

9380 

9663 

9743 

9828 

991 1 

9994 

♦077 

83 

523 

720139 

0242 

o325 

0407 

0490 

0573 

o655 

0738 

0821 

0903 

83 

526 

0986 

1068 

ii5i 

1233 

i3i6 

1398 

1481 

1 563 

1646 

1728 

82 

527 
52^ 

1811 

1893 

1975 

2o58 

2140 

2222 

23o5 

2387 

2469 

2552 

82 

2634 

2716 

2798 

2881 

2963 

3o45 

3127 

3209 

3291 

3374 

82 

529 

3456 

3538 

3620 

3702 

3/84 

3866 

3948 

4o3o 

4112 

4194 

82 

53o 

4276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

4931 

5oi3 

82 

53 1 

5095 

3176 

5258 

5340 

5422 

55o3 

5585 

5667 

5748 

583o 

82 

532 

5912 

5993 
6809 

6075 

6i56 

6238 

6320 

6401 

6483 

6564 

6646 

82 

533 

6727 

6890 

6972 

7053 

7'34 

7216 

7297 

7379 

7460 
8273 

81 

534 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8uo 

8191 

81 

535 

8354 

8435 

85i6 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

536 

9163 

9246 

9327 

9408 

9489 

9370 

9651 

9732 

9813 

9893 

81 

537 

*9974 

♦033 

oi36 

0217 

0298 

0378 

0459 

o54o 

0621 

0702 

81 

538 

730782 

o863 

0944 

1024 

1103 

1186 

1266 

1 347 

1428 

i5o8 

81 

539 

1589 

1669 

1750 

i83o 

I9II 

1991 

2072 

2l52 

2233 

23i3 

81 

540 

2394 

2474 

2555 

2635 

2715 

2796 

2876 

2956 

3o37 

3117 

80 

541 

3i97 

3278 

3358 

3438 

33i8 

3598 

3679 

3759 

3839 

3919 

80 

542 

3999 

4079 
48*0 

4160 

4240 

4320 

4400 

4480 

456o 

4640 

4720 

80  • 

543 

4800 

4960 

5o4o 

5l20 

5200 

5279 

5359 

5439 

5519 

80 

544 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6137 

6237 

63i7 

80 

545 

6397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7ii3 

80 

546 

7193 

7272 

7352 

743 1 
S225 

7511 

7590 

7670 

7749 

7829 

7908 

79 

547 

7987 

8067 

8146 

83o5 

8384 

8463 

8543 

8622 

8701 

79 

548 

8781 

8860 

8939 

9018 

9097 

9'77 

9256 

9335 

9414 

9493 

79 

549 

*9572 

965i 

9731 

9810 

98S9 

9968 

♦047 

0126 

0205 

0284 

79 

55o 

74  0363 

0442 

0321 

0600 

0678 

0737 

o836 

0915 

0904 

1073 

79 

55i 

Il52 

i23o 

i3o9 

1 388 

1467 

i546 

1624 

1703 

1782 

i860 

79 

552 

1939 

2018 

2096 

2175 

2254 

2332 

241 1 

2489 

2568 

2646 

79 

553 

2723 

2804 

2882 

2961 

3o39 
3823 

3ii8 

3196 

3273 

3353 

343 1 

7B 

554 

35io 

3588 

3667 

3745 

3902 

3980 

4038 

4i36 

4213 

78 

555 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

556 

5075 

5i53 

523i 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

557 

5855 

5933 

6011 

6089 

6167 

■  6245 

6323 

6401 

6479 

6556 

78 

558 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7179 

7236 

7334 

78 

559 

7412 

7489 

7567 

7643 

7722 

7800 

7878 

7953 

8o33 

8110 

78 

56o 

8188 

8266 

8343 

8421 

8498 

8576 

8653 

8731 

8808 

8885 

77 

56 1 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9304 

9582 

9639 

77 

562 

*9736 

9814 

9891 

9968 

♦045 

0123 

0200 

0277 

o334 

043 1 

77 

563 

75  o5o8 

o586 

o663 

0740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

564 

1279 

i356 

1433 

i5io 

1587 

1664 

1741 

1818 

1895 

1972 

77 

565 

2048 

2125 

2202 

2279 

2356 

2433 

2  509 

2586 

2663 

2740 

77 

566 

2816 

2893 

2970 

3o47 

3i23 

3200 

3277 

3353 

343o 

3506 

77 

567 

3583 

3660 

3736 

38i3 

3889 

3966 

4042 

4119 

4883 

4195 

4272 

77 

568 

4348 

4425 

45oi 

4578 

4654 

4730 

4807 

4960 

5o36 

76 

569 

5lI2 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

76 

570 

5875 

5951 

6027 

6io3 

6180 

6256 

6332 

6408 

6484 

656o 

76 

571 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

76 

572 

7396 

7472 

7548 
83o6 

7624 

7700 

7775 

785i 

7927 

8oo3 

8079 

76 

573 

8i35 

8230 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

76 

574 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

76 

575 

«9668 

9743 

9819 
0573 

9894 

9970 

♦045 

0121 

0196 

0272 

o347 

-5 

576 

76  0422 

0498 

0649 

0724 

0799 

0873 

0930 

1023 

IIOI 

73 

?3 

1 1 76 

1231 

i326 

1402 

1477 
2228 

l532 

1627 

1702 

1778 

1 853 

-3 

1928 

2oo3 

2078 

2i53 

23o3 

2378 

2433 

2329 

2604 

75 

579 

2679 

2754 

2829 

2904 

2978 

3o33 

3128 

32o3 

3278 

3353 

73 

N. 

0 

1    2 

3  1  4 

!    5 

6    7  1  8  1  9 

D. 

10 

LOGARITHMS  OF 

NUMBERS.     '     Table  I. 

N. 

0 

1 

2 

3 

4 

5 

6    7 

8 

9 

D. 

58o 

-6  3428 

35o3 

3578 

3653 

3727 

38o2 

3877 

3952 

4027 

4101 

75 

58 1 

4176 

425i 

4326 

4400 

4473 

455o 

4624 

4699 

4774 

4848 

75 

582 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

5445 

5520 

5594 

75 

583 

5669 

5743 

58i8 

5892 

5966 

6041 

6ii5 

6190 
6933 

6264 

6338 

74 

584 

64i3 

6487 

6562 

6636 

6710 

67S5 

6839 

7007 

7082 

74 

585 

7i56 

7230 

73o4 

7379 

7453 

7327 

7601 

7675 

7749 

7823 

74 

586 

7898 

7972 

8046 

8120 

S194 

8268 

8342 

8416 

8490 

8564 

74 

587 

8638 

87.2 

8786 

8860 

89J4 

9008 

9082 

9i56 

9230 

93o3 

74 

588 

*9377 

945 1 

9323 

9399 

9673 

9746 

9820 

9894 
o63i 

9968 

♦042 

74 

589 

770115 

0189 

0263 

o336 

0410 

0484 

0557 

0705 

0778 

74 

590 

o852 

0926 

0999 

1073 

1 146 

1220 

1293 

i367 

1440 

i5i4 

74 

591 

1387 

1661 

1734 

1808 

1S81 

1955 

2028 

2102 

2175 

2248 

73 

5o2 

2322 

2395 

2468 

2542 

26i5 

26S8 

2762 

2835 

2908 

2981 

73 

593 

3o55 

3 128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

594 

3786 

3860 

3933 

4006 

4079 

4i52 

4225 

4298 

4371 

4444 

73 

595 

4317 

4390 

4663 

4736 

4S09 

4882 

4955 

5028 

5ioo 

5173 

73 

596 

0246 

5319 

5392 

3465 

5538 

56io 

5683 

5736 

5829 

5902 

73 

597 

5974 

6047 

6120 

6193 

6263 

6338 

641 1 

6483 

6556 

6629 

73 

598 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

599 

7427 

7499 

7572 

7644 

7717 

77S9 

7862 

7934 

8006 

8079 

72 

600 

8131 

8224 

8296 

8368 

8441 

85i3 

8585 

8658 

S730 

8802 

72 

601 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

•  602 

*9596 

9669 

9741 

9813 

9885 

9937 

♦029 

010^ 

0173 

0245 

72 

6o3 

780317 

0389 

0461 

o533 

o6o3 

0677 

0749 

0821 

0893 

0965 

72 

604 

io37 

1109 

1181 

1253 

i324 

1396 

1468 

1340 

1612 

1684 

7' 

6o5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

606 

2473 

2544 

2616 

2688 

2739 

283i 

2902 

2974 

3046 

3117 

!'■ 

607 

3189 

3260 

333a 

34o3 

3473 

3546 

36i8 

36S9 

3761 

3832 

71 

608 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

44o3 

4475 

4546 

71 

609 

4617 

4689 

4760 

483 1 

4902 

4974 

5o45 

5ii6 

5187 

5239 

7' 

610 

533o 

5401 

5472 

5543 

56i5 

5686 

5757 

5828 

5899 

5970 

7« 

611 

6o4i 

6112 

6i83 

6254 

632  5 

6396 

6467 

6538 

6609 

6680 

7« 

612 

6731 

6822 

6893 

6964 

7035 

7106 

7'77 

7248 

7319 

7390 
8098 

71 

6i3 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7906 

8027 

71 

614 

8168 

8239 

83io 

838i 

8431 

8522 

8393 

8b63 

8734 

8S04 

7» 

6i5 

8875 

8946 

9016 

9087 

9'57 

9228 

9299 

9369 

9440 

9510 

71 

616 

*958i 

9631 

9722 

979a 

9863 

9933 

♦004 

0074 

0144 

0213 

70 

617 

79  0285 

o356 

0426 

0496 

0367 

0637 

0707 

0778 

0848 

0918 

70 

618 

0988 

1039 

1129 

1199 

1269 

i34o 

i4io 

1480 

i55o 

1620 

70 

619 

1691 

1761 

i83i 

1901 

1971 

205l 

2111 

2181 

2232 

2322 

70 

620 

2392 

2462 

2532 

2602 

2672 

2742 

2812 

2S82 

2952 

3022 

70 

621 

3092 

3i62 

323i 

33oi 

3371 

3441 

35ii 

3581 

365 1 

3721 

70 

622 

3790 

386o 

3930 

4000 

4070 

4i39 

4209 

4279 

4349 
5o45 

4418 

70 

623 

44H8 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5ii5 

70 

624 

5i85 

5254 

5324 

5393 

5463 

5532 

56o2 

5672 

5741 

58ii 

70 

625 

588o 

5949 

6019 
6713 

6088 

6i58 

6227 

6297 

6366 

6436 

65o5 

69 

626 

6374 

6644 

67S2 

6852 

6921 

6990 

7060 

7129 

7198 

^ 

627 

7268 

7337 

7406 

7475 

7545 

7614 

7683 

7732 

8443 

7821 

2c^° 

^9 

628 

7960 

8029 

8098 

8167 
8858 

8236 

83o5 

8374 

85i3 

8582 

69 

629 

865 1 

8720 

8789 

8927 

8996 

9065 

9134 

9203 

9272 

69 

63o 

9341 

9409 

9478 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

^ 

63 1 

80  0029 

0098 

0167 

0236 

o3o5 

0373 

0442 

o5ii 

o58o 

0648 

69 

632 

0717 

0786 

0834 

0923 

0992 

1061 

1129 

1198 

1266 

i335 

^9 

633 

1404 

1472 

i54i 

1609 
2295 

1678 

1747 

i8i5 

18S4 

1952 

2021 

69 

634 

2089 

2i58 

2226 

2363 

2432 

25oo 

2368 

2637 

2703 

69 

635 

2774 

2842 

2910 

2979 

3o47 

3n6 

3i84 

3252 

3321 

3389 

63 

636 

3457 

3525 

3594 

3662 

3730 

3798 

3S67 

3935 

4oo3 

4071 

68 

637 

4i39 

4208 

4276 

4344 

4412 

4480 

4548 

4616 

4685 

4753 

68 

638 

4821 

4889 

4957 

5o23 

5093 

5i6i 

5229 

5297 

5365 

5433 

68 

639 

55oi 

5569 

5637 

5705 

57-3 

5841 

5908 

5976 

6044 

6112 

68 

N. 

0 

1 

0 

3 

4 

5 

6    1 

8    9 

D. 

Table  I. 

LOGARITHMS  OF  NUMBERS.             11 

N.  ! 

0 

1    2 

3 

4 

5     6 

1 

8    9 

D. 

640 

806180 

6248 

63 1 6  63841 

645i 

65i9 

6587 

6655 

6723 

6790 

68 

641 

6858 

6926 

6994 

7061 

7129 

7'97 

7264 

7332 

7400 

8076 

7467 

68 

642 

7535 

7603 

7670 

7738 

7806 

7873 

794' 

8008 

8143 

68 

643 

8211 

8270 
8953 

8346 

8414 

84B1 

8549 

8616 

8684 

8751 

8818 

67 

644 

8886 

9021 

9088 

9136 

9223 

9290 

9358 

9425 

9492 

67 

645 

*956o 

9627 

9694 

9762 

9829 

9896 

9964 

♦o3i 

0098 

oi65 

67 

646 

810233 

o3oo 

o367 

0434 

o5oi 

0369 

o636 

0703 

0770 

0837 

67 

647 

0904 

0971 

1039 

1 106 

1173 

1240 

i3o7 

1374 

i44i 

i5o8 

67 

648 

1D75 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

64g 

2245 

23l2 

2379 

2445 

25l2 

2379 

2646 

2713 

2780 

2847 

67 

65o 

2913 

2980 

3047 

3iU 

3i8i 

3247 

33i4 

338i 

3448 

35i4 

67 

65 1 

3d8i 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

67 

652 

4248 

43 14 

438i 

4447 

45i4 

438 1 

4647 

4714 

4780 

4847 

67 

653 

4913 

4980 

5o46 

5ii3 

5179 
5843 

5246 

53i2 

5378 

5445 

55 1 1 

66 

654 

5578 

5644 

571 1 

5777 

5910 

5976 

6042 

6109 

6175 

66 

655 

6241 

63o8 

6374 

6440 

65o6 

6573 

6639 

6705 

6771 

6838 

66 

656 

6904 

6970 

7o36 

7102 

7169  ■ 

7235 

73oi 

7367 
8028 

J433 
8094 

7499 
8160 

66 

657 

7565 

7631 
6292 

7698 

7764 

783o 

7896 

7962 

66 

658 

8226 

83d8 

8424 

8490 

8556 

8622 

8688 

8734 

8820 

66 

659 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

*9544 

9610 

9676 

9741 

9807 

9873 

9939 

♦004 

0070 

oi36 

66 

661 

82  0201 

0267 

o333 

0399 

0464 

o33o 

0393 

0661 

0727 

0792 

66 

662 

o858 

0924 

0980 
1645 

io55 

1120 

1186 

I23I 

i3i7 

i382 

1448 

66-- 

663 

i5i4 

1 579 

1710 

1775 

1841 

.1906 

1972 

2037 

2io3 

65 

664 

■  2168 

2233 

2299 

2364 

243o 

2495 

256o 

2626 

2691 

2756 

65 

665 

2822 

2887 

2952 

3oi8 

3o83 

3148 

32i3 

3279 

3344 

3409 

65 

666 

3474 

3539 

36o5 

3670 

3735 

38oo 

3865 

3930 
458i 

3996 

4061 

65 

667 

4126 

4I9I 

4256 

4321 

4386 

4431 

45i6 

4646 

47II 

65" 

668 

4776 

4841 

4906 

4971 

5o36 

5ioi 

5i66 

523i 

5296 

536i 

65 

669 

5426 

5491 

5556 

5621 

5686 

5731 

58i5 

588o 

3945 

6010 

65 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

65 

671 

6723 

67S7 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

73o5 

65 

672 

7369 

7434 

7499 

7363 

7628 

7692 
8338 

7757 

7821 

7886 
853 1 

7951 

65 

673 

801  D 

8080 

8144 

8209 
8853 

8273 

8402 

8467 

8395 
92.39 

64 

674 

8660 

8724 

8789 

8918 

8982 

9046 

9111 

9175 

64 

675 

9304 

g368 

9432 

9497 
oi39 

9561 

9625 

9690 

9754 

9818 

9882 

64 

676 

«9947 

♦oil 

0075 

0204 

0268 

o332 

0396 

o46o 

o525 

64 

677 

83  0389 

o653 

0717 

0781 

0845 

0909 

I35o 

0973 

1037 

1102 

1166 

64 

678 

I23o 

1294 

1358 

1422 

i486 

1614 

1678 

1742 

1806 

64 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

23i7 

238i 

2445 

64 

680 

2  509 

2573 

2637 

2700 

2-64 

2828 

2892 

2956 

3020 

3o83 

64 

681 

3i47 

3211 

3275 

3338 

3402 

3466 

353o 

3593 
4230 

3657 

3721 

64 

682 

3784 

3848 

3912 

3975 

4o3g 
4675 

4io3 

4166 

4294 

4357 

64 

683 

4421 

4484 

4348 

4611 

4739 
5373 

4802 

4866 

4929 

4993 

64 

684 

5o56 

5l20 

5i83 

5247 

53io 

5437 

55oo 

5564 

5627 

63 

685 

5691 

5754 

5817 

588 1 

.  5944 

6007 

6071 

6i34 

6107 
683o 

6261 

63 

686 

6324 

6387 

6431 

65x4 

6377 

6641 

6704 

6767 

6894 

63 

687 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

688 

7652 

77i5 

7778 

7841 

7904 

7967 

8o3o 

8093 

8i56 

63 

689 

8219 

8282 

8343 

8408 

8471 

8334 

8397 

8660 

8723 

8786 

63 

6go 

8849 

8912 

8975 

9o38 

9101 

9164 

9227 

9289 

9352 

941 5 

63 

691 

*9478 

9341 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

♦043 

63 

692 

84oto6 

0169 

0232 

0294 

0357 

0420 

0482 

0343 

0608 

0671 

63 

693 

0733 

0796 

0859 

0921 

09S4 

1046 

1109 

II72 

1234 

1297 

63 

694 

i359 

1422 

J  485 

1047 

1610 

1672 

1735 

•797 

i860 

1922 

63 

695 

1985 

2047  ;  2110 

2172 

2235 

2297 

236o 

2422 

2484 

2547 

63 

696 

2609 

2672  !  2734 

2796 

2839 

2921 

2983 

3  046 

3io8 

3170 

62 

697 

3233 

3295  !  3357 

3420 

3482 

3344 

36o6 

3669 

3731 

3793 

62 

698 

3855 

3918  1  3980 

4042 

4104 

4166 

4229 

4291 

4353 

44i5 

62 

699 

4477 

4339  1  4601 

4664 

4726  . 

4788 

4830 

4912 

4974 

5o36 

62 

N. 

0 

1  1  1  2 

1  3  i  4 

1   5 

6  1  7    8 

9  1  D.  1 

12 

LOGARITHMS  OF  NUMBERS.         Table  I. 

N. 

0 

1 

2 

3 

4 

5 

6 

1 

8 

9 

D. 

700 

84  5098 

5i6o 

3222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

6j 

701 

5718 

5780 

5842 

5oo4 
6528 

5966 

6028 

6090 

6i5i 

62J3 

6275 

6a 

702 

6337 

5399 

6461 

6385 

6646 

6708 

6770 

6832 

6894 

6a 

7o3 

6955 

7017 

7079 

7141 

7202 

7264 

7826 

7888 

7449 

75.1 

62 

704 

7673 

7684 

7696 

7738 

7819 

7881 

7943 

8004 

8066 

8128 

62 

705 

8189 

825i 

83i2 

8874 

8435 

8497 

8559 

8620 

8682 

8743 

62 

706 

8SoD 

8866 

8028 

8989 

9o5i 

91 1 2 

9174 

9235 

9?97 

9358 

61 

7o8 

9410 
85oo33 

9481 

9642 

9604 

9665 

9726 

9788 

9849 

991 1 

o585 

61 

0095 

oi56 

0217 

0279 

o34o 

o4oi 

0462 

0324 

61 

709 

0646 

0707 

0769 

0880 

0891 

0952 

1014 

1075 

u36 

1 197 

61 

710 

1258 

1820 

i38i 

1442 

i5o3 

i564 

1625 

1686 

1747 
2358 

1809 

61 

711 

1870 

1981 

1992 

2o53 

2114 

2175 

2236 

:297 

2419 

61 

712 

2480 

234 1 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

8029 

61 

7.3 

8090 

3i5o 

3211 

8272 

8333 

8894 

3455 

35i6 

3377 

3637 

61 

7J4 

3698 

3759 

8820 

388i 

8941 

4002 

4063 

4124 

4i85 

4245 

61 

7.5 

43o6 

4867 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

716 

4913 

4974 

5o84 

5095 

5i36 

5216 

5277 

5337 

5398 

5459 

61 

7«7 

5519 

558o 

5640 

5701 

5761 

5822 

5882 

5948 

6008 

6064 

61 

718 

6124 

6i85 

6245 

6806 

6366 

6427 

6487 

6348 

6608 

6668 

60 

719 

6729 

6789 

685o 

6910 

6970 

7081 

7091 

7i52 

7212 

7272 

60 

720 

7332 

7898 

7453 

75i3 

7574 

7684 

7694 

7755 

7815 

7875 

60 

721 

7935 

7995 

8o56 

8116 

8176 

8286 

8297 

8357 
8958 

8417 

8477 

60 

722 

8D37 
9'3» 

8397 

8657 

8718 

8778 

8838 

8898 

9018 

9078 

60 

723 

9198 

9258 

9818 

9879 

9489 

9499 
0098 

9359 

96.0 

9679 

60 

724 

♦  9739 

9799 

9859 

9918 

9978 

♦o38 

oi58 

0218 

0278 

60 

725 

86o338 

0898 

0458 

o5i8 

0578 

0637 

0697 

0757 

0817 

0877 

60 

726 

0987 

0996 

io56 

1116 

1176 

1236 

1295 

i355 

I4i5 

1475 

60 

727 

1 534 

1394 

1 654 

1714 

1773 

1 883 

1898 

1952 

2012 

2072 

60 

728 

2i3i 

2101 

225l 

23jo 

2870 

2480 

2489 

2349 

2608 

2668 

60 

729 

2728 

2787 

2847 

2906 

2966 

3023 

3o85 

3i44 

3204 

8268 

60 

730 

3323 

3382 

3442 

35oi 

356 1 

8620 

368o 

3789 

3799 

3858 

59 

73 1 

3917 

3977 

4086 

4096 

4i55 

4214 

4274 

4833 

4802 

4452 

59 

732 

45ii 

4570 

4680 

4689 

4748 

4808 

4867 

4936 

4985 

5o45 

5o 

733 

5io4 

5i68 

5222 

5282 

5341 

5400 

5459 

5319 

5578 

5687 

59 

734 

5696 

5755 

58i4 

5874 

5983 

5992 

6o5i 

6jio 

6169 

6228 

59 

735 

6287 

6846 

64o5 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

736 

6878 

6937 
7526 

7D^5 

7055 

7114 

7173 

7282 

7291 

735o 

7409 

59 

737 

7467 

7644 

7708 

835o 

7821 

7880 

7989 

7998 

59 

738 

8o56 

8ii5 

8174 

8288 

8292 

8409 

8468 

8527 

8586 

59 

739 

8644 

8703 

8762 

8821 

8879 

8988 

8997 

9o56 

9114 

9178 

4 

740 

9282 

9290 

9849 
9935 

9408 

9466 

9523 

9584 

9642 

9701 

9760 

59 

741 

*98i8 

9877 

9994 

♦o58 

OIII 

0170 

0228 

0287 

0845 

59 

742 

87  0404 

0462 

0321 

0379 

o638 

0696 

0755 

0818 

0872 

0980 

58 

743 

0989 
1373 

1047 

1 106 

1164 

1223 

1281 

1339 

1398 
J981 

1456 

I3l5 

58 

744 

1681 

1690 

1748 

1806 

i865 

1923 

2040 

2098 

58 

745 

2i56 

22l5 

2273 

233i 

2389 

2448 

25o6 

2564 

2622 

2681 

58 

746 

2789 

2797 

2855 

2918 

2972 

3o8o 

3o88 

3i46 

3204 

3262 

58 

747 

3321 

8379 

8437 

3495 

3358 

36u 

8669 

3727 

3785 

3844 

58 

748 

8902 

8960 

4018 

4076 

4i34 

4192 

425o 

43o8 

4366 

4424 

58 

749 

4482 

4340 

4598 

4656 

4714 

4772 

4880 

4888 

4945 

5oo3 

58 

75o 

5o6i 

5119 

5698 

5i77 

5235 

5j93 

535i 

5409 

5466 

5524 

5582 

58 

75i 

5640 

5756 

58i3 

5871 

5929 
6307 

5987 

6045 

6102 

6160 

58 

,752 

6218 

6276 

6333 

6891 

6449 

6564 

6622 

6680 

6787 

58 

753 

6795 

6853 

6910 

6968 
7544 

7026 

7088 

7141 

7199 

7256 

7314 

58 

754 

7871 

7429 

7487 

7602 

7659 

77«7 

7774 

7832 

7889 

58 

755 

7947 

8004 

8062 

8119 

8177 

8284 

8292 

8849 

8407 

8464 

57 

756 

8322 

8579 

8687 

8694 

8752 

8809 
9888 

8866 

8924 

8981 

9089 

57 

757 

9096 

9153 

9211 

9268 

9825 

9440 

9497 

9355 

9612 

57 

758 

«  9669 

9726 

9784 

9841 

9898 

9956 

♦oi3 

0070 

0127 

oi85 

57 

759 

88  0242 

0299 

o356 

0418 

0471 

0528 

o585 

0642 

0699 

0756 

57 

N. 

0 

1 

0 

3 

4 

5 

6 

7 

8 

9 

D. 

Table  I. 

LOGARITHMS  OF  NUMBERS. 

18 

N. 
760 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

57 

880814 

0871 

0928 

0985 

1042 

1099 

ii56 

I2l3 

1271 

1828 

76. 

i385 

1442 

1499 

id56 

i6i3 

1670 

1727 

1784 

1841 

1898 

37 

762 

1955 

2525 

2012 

2069 

2126 

2i83 

2?4o 

2297 

•2354 

2411 

2468 

57 

763 

258i 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3o37 

57 

764 

3093 

3i5o 

3207 

3264 

3321 

3377 

3434 

3491 

3348 

36o3 

57 

76,5 

366 1 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4ii5 

4172 

57 

766 

4229 

4285 

4342 

4399 

4455 

4312 

4369 

4625 

4682 

4739 

57 

767 

4793 

4852 

4909 

4963 

5o2  2 

5078- 

5i35 

5192 

3248 

53o3 

57 

768 

5361 

5418 

5474 

533 1 

5587 

5644 

5700 

5757 

58i3 

5870 

a 

769 

5926 

5g83 

6039 

6096 

6i52 

6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942  6998 

56 

-771 

7054 

71M 

7167 

7223 

7280 

7336 

7392 

7449 

73o5  7561 

56 

772 

7617 
^179 

7674 

773o 

7786 

7842 
8404 

7898 
8460 

7935 

8011 

8067  8123 

56 

773 

8236 

8292 

8348 

83i6 

8573 

8629  8685 

56 

774 

8741 

8797 

8853 

8909 

8963 

9021 

9077 

9134 

9190 

9246 

56 

775 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

56 

776 

*9862 

9918 

9974 

♦o3o 

0086 

0141 

0197 

0253 

o3o9 

0365 

56 

777 

890421 

0477 

o533 

o58q 

0643 

0700 

0736 

0812 

0868 

0924 

56 

778 

0980 

io35 

1091 

1 147 

I203 

1259 

i3i4 

1370 

1426 

1482 

56 

779 

1337 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983  2039 

56 

780 

2095 

2i5o 

2206 

2262 

23i7 

2373 

2429 

2484 

2540  2595 

56 

78. 

265i 

2707 

2762 

2818 

2873 

2929 

2985 

3o4o 

3096  3l3I 

56 

782 

3207 

3262 

33i8 

3373 

3429 

3484 

3540 

3595 

3631  3706 

56 

783 

3762 

3817 

3873 

3928 

3984 

4039 
4593 

4094 

4i5o 

42o5  4261 

55 

784 

43 16 

4371 

4427 

4482 

4538 

4648 

4704 

4759 

4814 

55 

785 

4870 

4925 

4980 

5o36 

5091 

5i46 

5201 

5257 

53i2 

5367 

55 

786 

5423 

5478 

5d33 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

55 

787 

5o75 

6o3o 

6o85 

6140 

6195 

6231 

63o6 

636i 

6416 

6471 

55 

788 

6526 

658i 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

789 

7077 

7i32 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

7627 

7682 

7737 

7792 

7847 

7902 
845i 

7957 

8012 

8067 

8122 

55 

79' 

8176 

823i 

8286 

8341 

8396 

83o6 

856i 

86i5 

8670 

55 

792 

8725 

8780 

8835 

8890 
9437 

8944 

8999 
9547 

9054 

9109 

9164 

9218 

55 

793 

9273 

9328 

9383 

9492 

9602 

9656 

97" 

9766 

55 

794 

#9821 

9875 

9930 

9985 

♦o39 

0094 

0149 

0203 

0258 

03l2 

55 

795 

90  0367 

0422 

0476 

o53i 

o586 

0640 

0695 

0749 

0804 

0859 

55 

7g6 

0913 

0968 

1022 

1077 

ii3i 

1 186 

1240 

1293 

1 349 

1404 

55 

797 
798 

1458 

lDl3 

1 567 

1622 

1676 

1 73 1 

1785 

1840 

1894 
2438 

1948 

54 

2oo3 

2o57 

2112 

2166 

2221 

2275 

2329 
2873 

2384 

2492 
3o36 

54 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2927 

2981 

54 

800 

3090 
3633 

3i44 

3199 

3:53 

3307 

336i 

3416 

3470 

3524 

3578 

54 

801 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

8o3 

4716 

4770 

4824 

4878 

4932 

4986 

5o4o 

5094 

5i48 

5202 

54 

804 

5256 

53io 

5364 

5418 

5472 

5326 

558o 

5634 

5688 

5742 

54 

8o5 

5796 
6335 

585o 

5904 

5958 

6012 

6066 

6119 

6173 

6227  6281 

54 

806 

6389 

6443 

6407 
7o35 

655i 

6604 

6658 

6712 

6766 

6820 

54 

807 

6874 

6927 

6981 

7089 

7143 

7196 
7734 
8270 

725o 

7304 

7358 

54 

808 

741 1 

7465 

7D19 

7573 

7626 

7680 

7787 

7841 
8378 

7805 
843 1 

54 

809 

7949 

8002 

8o56 

8110 

8i63 

8217 

8324 

54 

810 

8485 

8539 

85g2 

8646 

8609 
9233 

8753 

8807 

8860 

8914 

8967 

54 

8it 

9021 

9074 

9128 

9181 

9280 
9823 

9342 

9396 

9449 

9303 

54 

812 

*9556 

9610 

9663 

9716 

9770 

9877 

9930 

9984 

♦037 

53 

8i3 

91  0091 

0144 

0197 

025l 

o3o4 

o358 

0411 

0464 

03l8 

0571 

53 

814 

0624 

0678 

0731 

0784 

o838 

0891 

og44 

0998 

io5i 

1104 

53 

8i5 

ii58 

1211 

1264 

i3i7 

1371 

1424 

1477 

i53o 

1 584 

1637 

53 

816 

1690 

1743 

1797 

i85o 

1903 

1956 

2009 

2o63 

2116 

2169 

53 

817 

2222 

2275 

2328 

238i 

2435 

2488 

2541 

2594 

2647 

2700 

53 

818 

2753 

2806 

2859 

2913 

2966 

3019 

3072 

3i25 

3178 

323i 

53 

819 

3284 

3337 

3390 

3443 

3496 

3549 

36o2 

3655 

3708 

3761 

53 
D. 

N. 

0 

1 

2 

3 

4 

5 

6    7 

8  1  9 



14 

LOGARITHMS  OF  NUMBERS.         TABtB  L 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

820 

9i38i4 

3867 

3920 

3973 

4026 

4070 
4608 

4i32 

4184 

4237 

4290 

53 

821 

4343 

4396 

4449 

45o2 

4555 

4660 

4713 

4766 

4819 

53 

822 

4872 

4925 

•4977 

5o3o 

5o&3 

5i36 

3189 

5241 

5294 

5347 

53 

823 

5400 

5453 

55o5 

5558 

56ii 

5664 

3716 

3769 

5822 

5875 

53 

824 

5927 

5980 

6o33 

6o85 

6i38 

6191 

6243 

6296 

6349 

-6401 

53 

825 

6454 

65o7 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

826 

6980 

7033 

708D 

7i38 

7190 

7243 

7295 

7348 

7400 

7453 

53 

827 

7506 
8o3o 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

32 

828 

8o83 

8i35 

8188 

8240 

8293 

8345 

8397 

84  5o 

8302 

52 

829 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

83o 

9078 

gi3o 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

52 

83 1 

*gboi 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

♦019 

0071 

52 

832 

92  0123 

0176 

0228 

0280 

o332 

o384 

0436 

0489 

0341 

0593 

52 

833 

0645 

0697 

0749 

0801 

o853 

0906 

0958 

1010 

1062 

1114 

52 

834 

1166 

1218 

1270 

l322 

1374 

1426 

1478 

i53o 

i582 

i634 

52 

835 

1686 

1738 

1790 

1842 

1894  . 

1946 

1998 

2o5o 

2102 

2i54 

52 

836 

2206 

2258 

23l0 

2362 

2414 

2466 

23l8 

2570 

2622 

2674 

52 

837 

2725 

2777 

2829 

2881 

2933 

2985 

3o37 

3089 

3i4o 

3192 

52 

838 

3244 

3296 

3348 

3399 

345i 

35o3 

3555 

3607 

3658 

3710 

52 

839 

3762 

38i4 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

32 

840 

4279 

433i 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

52 

84 1 

479'^ 

4848 

4899 

4951 

5oo3 

5o54 

5 106 

5i57 

5209 

5201 

52 

842 

5312 

5364 

54i3 

5467 

55i8 

5570 

5621 

5673 

5725 

5776 

52 

843 

5828 

5879 

5931 

5982 

6o34 

6o85 

6137 

6188 

6240 

6291 

5i 

844 

6342 

6394 

6445 

6497 

6548 

6600 

665 1 

6702 

6734 

68o5 

5i 

845 

6857 

6908 

6959 

7473 

701 1 

7062 

7114 

7165 

7216 

7268 

7319 

5i 

846 

7370 

7422 

7524 

7576 

7627 

767S 

7730 
8242 

7781 

7832 

5i 

847 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8293 

8343 

31 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

88o5 

8857 

5i 

849 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

5i 

S3o 

9419 

9470 

9521 

9372 

9^23 

9674 

9725 

9776 

9827 

9879 

5i 

801 

*  9930 

99«i 

♦032 

oo83 

01 34 

0185 

0236 

0287 

o338 

0389 

5i 

852 

93  0440 

0491 

o542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

5i 

853 

0949 

1000 

io5i 

1102 

ii53 

1204 

1254 

i3o5 

i356 

1407 

5i 

854 

1458 

i5o9 

i56o 

1610 

1661 

1712 

1763 

1814 

1863 

1915 

5i 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

5i 

856 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

5i 

857 

2981 

3o3i 

3o82 

3i33 

3i83 

3234 

3285 

3335 

3386 

3437 

5i 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

5i 

839 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

5i 

860 

4498 

4549 

4399 

465o 

4700 

4751 

4801 

4852 

4902 

4953 

5o 

861 

5oo3 

5o54 

5io4 

5i54 

52o5 

5255 

53o6 

5356 

5406 

5457 

5o 

862 

55o7 

5558 

56o8 

5658 

5709 

5759 

5809 

586o 

5910 

5960 

5o 

863 

60H 

6061 

6111 

6162 

6212 

6262 

63i3 

6363 

64 1 3 

6463 

5o 

864 

65i4 

6564 

6614 

6665 

6713 

6765 

68i5 

6865 

6916 

6966 

5o 

865 

7016 

7066 

7117 

7.67 

7217 

7267 

7317 

7367 

7418 

7468 

5o 

866 

75i8 

7068 

7618 

7668 

7718 

77('9 

7819 

7869 

7919 

7969 

5o 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

5o 

868 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

5o 

86q 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

5o 

870 

9519 
940018 

9569 

9610 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

5o 

871 

0068 

0118 

0168 

0218 

0267 

o3i7 

o367 

0417 

0467 

5o 

872 

o5i6 

o566 

0616 

0666 

0716 

0763 

o8i3 

o865 

0915 

0964 

5o 

873 

1014 

1064 

1114 

ii63 

12l3 

1263 

i3i3 

i362 

1412 

1462 

5o 

874 

i5ii 

i56i 

i6u 

1660 

1710 

1760 

1809 

1839 

1909 

1958 

5o 

875 

2008 

2o58 

2107 

2157 

2207 

2256 

23o6 

2355 

24o5 

2455 

5o 

S76 

2304 

2554  '  26o3 

2653 

2702 

2752 

2801 

285i 

2901 
3396 

2950 

5o 

^11 

3ooo 

3o49 

3099 

3i48 

3198 

3247 

3297 

3346 

3445 

49 

878 

3495 

3544 

3593 

3643 

3692 

3742 

3701 

3841 

3890 

3939 
4433 

49 

870 

3989 

4o38 

4088 

4i37 

4186 

4236 

4285  I4335 

4384 

49 

N. 

0 

1 

0 

3 

4 

5  1  6  j  7  1  8 

9 

D. 

Table  I. 

LOGARITHMS  OF  NUMBERS. 

16 

N.  1 

0 

1    2  1  3  1  4 

5 

6 

7 

8 

9 

D. 

49 

880 

944483 

4532 

458 1 

463 1  j 

4680 

4729 

4779 

4828 

4877 

4927 

881 

4976 

5023 

5o74 

5i24 

5173 

5222 

5272 

5321 

5370 

5419 

49 

882 

5469 

55i8 

5567 

56i6 

5665 

5713 

5764 

58i3 

5862 

5912 

49 

883 

3961 

6010 

6059 

6108 

6157 

6207 

6256 

63o5 

6354 

6403 

49 

884 

6432 

6301 

655i  1  6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

885 

6943 

6992 

7041  7090 

7140 

7189 

7238 

7287 

7336 

7385 

49 

886 

7434 

7483 

7532 

758 1 

763o 

7(^79 

7728 

7777 

7826 

7875 

49 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

83i5 

8364 

49 

888 

84 1 3 

8462 

85ii 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

49 

889 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

890 

9390 

9439 

9488 

9336 

9585 

9634 

9683 

973 1 

9780 

9829 

49 

891 

*98-8 

9926 

9975 

♦024 

0073 

0121 

0170 

0219 

0267 

o3i6 

49 

892 

93  0365 

0414 

0462 

o5ii 

o56o 

0608 

0657 

0706 

0754 

o8o3 

49 

893 

o85i 

0900 
1 386 

0949 

0997 

1046 

1093 

1143 

1192 

1240 

1289 

49 

894 

i338 

i43d 

1483 

i532  • 

i58o 

1629 

1677 

1726 

1773 

49 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2i63 

2211 

2260 

48 

896 

2  3o8 

2356 

2403 

2453 

2502 

2330 

2309 
3o8J 

2647 

2606 

2-44 

48 

897 

2792 

2841 

2889 
3373 

2938 

2986 

3o34 

3i3i 

3 180 

3228 

48 

898 

32-6 

3325 

3421 

3470 

35i8 

3566 

36j5 

3663 

37 1 1 

48 

899 

3760 

38o8 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

4243 

4291 

4339 

4387 

4435 

4484 

4532 

458o 

4628 

4677 

48 

901 

4723 

4773 

4821 

4869 

4918 

4966 

3014 

5o62 

5iio 

5i58 

43 

902 

5207 

5255 

53o3 

535i 

5399 

5447 

5493 

5543 

5592 

5640 

48 

903 

5688 

5736 

5784 

5832 

588o 

5928 

5976 

6024 

6072 

6120 

48 

904 

6168 

6ii6 

6265 

63i3 

636i 

6409 

6457 

65o5 

6553 

6601 

48 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7o32 

7080 

43 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

75i2 

7559 

48 

907 

7607 

7655 

77o3 

7731 

7799 

7847 

7894 

7942 

7990 

8o38 

48 

908 

80&6 

8i34 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

85i6 

48 

909 

8564 

8612 

8659 

8707 

8755 

88o3 

885o 

8898 

8946 

8994 

48 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

947" 

48 

911 

9318 

9566 

9614 

9661 

9709 

9737 

9804 

9852 

9900 

9947 

48 

912 

*9995 

♦042 

0090 

oi38 

oi85 

0233 

0280 

o328 

0376 

0423 

48 

9.3 

96  047 1 

o5i8 

o566 

o6i3 

0661 

0709 

0756 

0804 

o85i 

0S99 

48 

914 

0946 

0994 

1041 

1089 

ii36 

1184 

I23l 

1279 

i326 

1374 

47 

9i5 

1421 

1469 
1943 

i5i6 

1 563 

1611 

i658 

1706 

1753 

1801 

1848 

47 

916 

1895 

1990 

2o38 

2o85 

2l32 

2180 

2227 

2273 

2322 

47 

9"7 

2369 
2843 

2417 

24t)4 

25ll 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

2890 

2937 

2985 

3o32 

3079 

3126 

3174 

3221 

3263 

47 

919 

33i6 

3363 

3410 

3437 

35o4 

3552 

3599 

3646 

3693 

3741 

47 

920 

37S8 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4i63 

4212 

47 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

922 

473 1 

477S 

4825 

4872 

4919 

4966 

5oi3 

5o6i 

5io8 

5i55 

47 

923 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

553 1 

5578 

5625 

47 

924 

56-2 

5719 

3766 

58i3 

5860 

5907 

5954 

6001 

6048 

6095 

47 

923 

6142 

6i8q 

6236 

6283 

6329 

6376 

6423 

6470 

6317 

6564 

47 

926 

661 1 

6658 

6705 

6752 

6799 

6845 

6892 

6930 

6986 

7033 

47 

927 

7080 

7127 

7173 

7220 

7267 

73i4 

736i 

7408 

7454 

7301 

47 

928 

7348 

7593 

7642 

7688 

7735 

7782 

8296 

7875 

SS 

7969 

47 

929 

8016 

8062 

8109 

81 56 

8203 

8249 

8343 

8436 

47 

93o 

8483 

853o 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8oo3 
9369 

47 

93. 

8930 

8996 

9043 

9090 

9i36 

9183 

9229 

9276 

9323 

47 

932 

9416 

9463 

9509 

9556 

9602 

9649 

9693 

9742 

97S9 

9835 

47 

933 

^9882 

9928 

9973 

♦021 

0068 

0114 

0161 

0207 

0254 

o3oo 

47 

934 

97  0^47 

0393 

0440 

0486 

o533 

0579 

0626 

0672 

0719 

0765 

46 

935 

0812 

o858 

0904 
1369 

0951 

0997 

1044 

1090 

ii37 

1183 

1229 
1693 

46 

936 

1276 

l322 

I4i5 

1461 

i5o8 

1 554 

1601 

1647 

46 

937 

1740 

1786  i  1 832 

1879 

1923 
2388 

J971 

2018 

2064 

2110 

2137 

46 

933 

2203 

2249  2295 

2342 

2434 

2481 

2527 

2373 

2619 

46 

939 

2666 

2712  '.  2758 

2804 

285i 

2897 

2943 

2989 

3o35 

3o82 

46 

N. 

0 

1  1    2 

1  « 

4 

1   '^ 

6 

7  1  8 

9 

D. 

16 

LOGARITHMS  OF  NUMBERS.         Table  L 

N. 

0 

1 

2 

3 

4 

5 

6    7 

8 

9 

D. 

940 

973128 

3174 

3220 

3266 

33i3 

3359 

34o5 

345i 

3497 

3543 

46 

941 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3918 

3939 

400  5 

46 

942 

4031 

4097 

4143 

4189 

4235 

4281 

4327 

4874 

4420 

4466 

46 

943 

4312 

4558 

4604 

465o 

4696 

4742 

4788 

4834 

4880 

4926 

46 

944 

4972 

5oi8 

5o64 

5iio 

5i56 

5202 

5248 

5294 

5340 

5386 

46 

945 

5432 

5478 

5524 

5570 

56i6 

5662 

5707 

5753 

5799 

5845 

46 

946 

5891 

5937 

5983 

6029 

6488 

6075 

6121 

6167 

6212 

6258 

6804 

46 

947 

63  30 

6396 

6442 

6533 

6379 

6625 

6671 

6717 

6768 

46 

948 

6808 

6834 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

949 

7266 

73i2 

7358 

7403 

7449 

7495 

7541 

7586 

7682 

7678 

46 

950 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8i35 

46 

901 

8181 

8226 

8272 

83i7 

8363 

8409 

8454 

85oo 

8546 

8591 

46 

932 

8637 

8683 

8728 

8774 

8819 

8865 

891 1 

8956 

9002 

9047 

46 

953 

9093 

9i38 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9508 

46 

954 

9348 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

955 

98  ooo3 

0049 

0094 

0140 

oi85 

023l 

0276 

0822 

0867 

0412 

45 

936 

0458 

o5o3 

o549 

0594 

0640 

o685 

0780 

0776 

0821 

0867 

45 

937 

0912 

0957 

ioo3 

1048 

1093 

1 1 39 

1184 

1229 

1275 

1820 

45 

938 

i366 

1411 

1456 

i5oi 

1 547 

1392 

1687 

1683 

1728 

1778 

45 

939 

1819 

1864 

1909 

1954 

2000 

2043 

2090 

2i35 

2181 

2226 

45 

960 

2271 

23i6 

2362 

2407 

2452 

2497 

2543 

2588 

2683 

2678 

45 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

8040 

3o85 

3i3o 

45 

962 

8175 

3220 

3265 

33io 

3356 

3401 

8446 

8491 

3536 

358i 

45 

963 

3626 

3671 

3716 

3762 

3807 

3852 

8897 

8942 

3987 

4082 

45 

964 

4077 

4122 

4167 

4212 

4257 

43o2 

4847 

4392 

4437 

4482 

45 

965 

4527 

4572 

4617 

4662 

4707 

4732 

4797 

4842 

4887 

4932 

45 

966 

4977 

5022 

5067 

5ll2 

5i57 

5202 

5247 

5292 

5337 

5382 

45 

967 

5426 

5471 

55i6 

556i 

56o6 

565i 

5696 

5741 

5786 

583o 

45 

968 

5875 

5920 

5965 

6010 

6o55 

6100 

6144 

6189 

6284 

6279 

45 

969 

6324 

6J69 

641 3 

6458 

65o3 

6548 

6598 

6687 

668:. 

6727 

45 

970 

6772 

6817 

6861 

6906 
7.353 

6951 
7398 

6996 

7040 

7085 

7180 

7175 

45 

971 

7219 

7264 

7309 

7443 

7488 

7582 

7577 

7622 
8068 

45 

972 

7666 

77II 

7756 

7800 

7845 

7890 
^336 

7984 

7979 

8024 

45 

973 

8i!3 

8157 

8202 

8247 

8291 

838i 

8423 

8470 

85i4 

45 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

975 

9005 

9049 

9004 

9i38 

9183 

9227 

9272 

9816 

9361 

94o5 

45 

976 

94  30 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

44 

977 

*9895 

9939 

o383 

9983 

♦028 

0072 

0117 

0161 

0206 

025o 

0294 

0788 

44 

978 

99  0339 

0428 

0472 

o5i6 

036l 

o6o5 

o65o 

0694 

44 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1187 

1182 

44 

980 

1226 

1270 

i3i5 

i339 

i4o3 

1448 

1492 

1 586 

i58o 

1625 

44 

98. 

1669 

17.3 

1758 

1802 

1846 

1890 

1935 

J  979 

2023 

2067 

44 

982 

21 II 

2 1 56 

2200 

2244 

2288 

2333 

2877 

2421 

2465 

2  509 

44 

983 

2554 

2398 
3o39 

2642 

2686 

2730 

2774 

2819 

2863 

2907 
8848 

2951 

44 

984 

2995 

3o83 

3127 

3172 

3216 

3260 

33o4 

8892 

44 

985 

3436 

3480 

3524 

3568 

36i3 

3657 

8701 

3745 

3789 

3833 

44 

986 

3877 

3921 

3965 

4009 

4o53 

4097 

4141 

4i85 

4229 

4278 

44 

987 

4317 

436i 

44o5 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

988 

4757 

4801 

4845 

4889 
5328 

4933 
5372 

4977 

502I 

5o65 

5io8 

5i52 

44 

989 

5196 

5240 

5284 

5416 

5460 

55o4 

5547 

5591 

44 

990 

5635 

5679 

5723 

5767 

58ii 

5854 

5898 

5942 
638o 

5986 

6o3o 

44 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6424 

6468 

44 

992 

65i2 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

993 

6949 

6993 

7037 

7080 

7«24 

7168 

7212 

7255 

7299 
7736 

7343 

44 

994 

7386 

743o 

7474 

7517 

7361 

7605 

7648 

7692 

7779 

44 

995 

7823 

7867 

7910 
8347 

7954 
8390 

7908 

804 1 

8o85 

8129 

8172 

8216 

44 

996 

8259 

83a3 

8434 

8477 

8521 

8564 

8608 

■8652 

44 

997 

8693 

8739 

8782 

8826 

8869 

8913 
9348 

8956 

9000 

9043 

9087 

44 

998 

9j3i 

9174 

9218 

9261 

9803 

9892 

9435 

9479 

9522 

44 

999 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9918 

9957  43  1 

N. 

0 

1 

0 

3 

4 

5 

6 

1 

8 

9 

M 

TABLE    II. 


lOGARITHMIC   SINES    AND   TANGENTS, 


EVERY  DEGREE  AND  MI.VUTE  OF  THE  QUADRANT. 


If  tlie  log-aritlims  of  the  values  in  Table  III.  be  eacli  increased  by  lo,  the  resulta 
will  be  the  values  of  this  table. 

The  logarithmic  Secants  and  Cosecants  are  not  given.  They  may  be  readily  ob- 
tained, as  follows: — Subtract  the  logarithmic  Cosine  from  20,  and  the  remainder 
will  be  the  logarithmic  Secant ;  subtract  the  logarithmic  Sine  from  ao,  and  the 
ronaaiader  will  be  the  lojarithmi*  Cosecant. 


18 

LOaARITHMIC  SINES, 

TANGENTS,  ETC 

Table 

TH 

0° 

179°  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

Inf.  Neg. 

1 0  -  000000 

Inf.  Neg. 

Infinite. 

60 

I 

6-463726 

501717 
293485 

000000 

00 

6-468726 

501717 

13.586274 

U 

3 

764736 

000000 

00 

764756 

298488 

285244 

3 

940847 

208231 

000000 

00 

940847 

208281 

059 1 53 

57 

4 

7-065786 

i6i5i7 
131968 
111375 

000000 

00 

7-065786 

i6i5i7 

12.984214 

56 

5 

162696 

000000 

00 

162696 

181969 

087804 

55 

6 

241877 

9-999999 

01 

241878 

111378 

758122 

54 

I 

308824 

96653 

999999 

01 

308825 

99653 

§3254 

691175 
633 1 83 

53 

3668 1 6 

85254 

999999 

01 

366817 

52 

9 

417968 

76263 

999999 

01 

417970 

76268 

682080 

5i 

10 

463726 

68988 

999998 

01 

468727 

68988 

536273 

5o 

II 

7.5o5ii8 

62981 

9.999998 

01 

7.5o5i2o 

62981 

12.494880 

49 

12 

542906 

57936 

999997 

01 

542909 

57988 

457091 

48 

i3 

577668 

53641 

993997 

01 

577-572 

58642 

422828 

47 

i4 

609853 

49938 

999996 

0: 

609857 

49989 
46715 

890148 

46 

i5 

539816 

46714 

999996 

01 

689820 

36oi8o 

45 

i6 

([67845 

43881 

999995 

01 

667849 

43882 

382i5i 

44 

17 

694173 

41372 

999995 

01 

694179 
719003 

41878 

3o582i 

43 

i8 

718997 

39135 

999994 

01 

89186 

280997 

42 

'9 

742478 

37127 

999993 

01 

742484 

37128 

257316 

41 

20 

764734 

353i5 

999993 

01 

764761 

35i36 

235289 

40 

21 

7.785943 
806146 

33672 

9-999992 

01 

7.785951 

38673 

12.214049 

39 

22 

32175 

999991 

01 

8061 55 

82176 

193845 

38 

23 

825451 

3o8o5 

999990 

01 

825460 

3o8o6 

174540' 

37 

24 

843984 

29547 

999989 

02 

848944 

29549 

i56o56 

36 

25 

861662 

28388 

999989 

02 

861674 

28890 

188826 

35 

26 

878695 

27317 

999988 

02 

878708 

27818 

121292 

34 

27 

895085 

26323 

999987 

02 

895099 

26825 

104901 

38 

28 

910879 

25399 

999986 

02 

910894 
926184 

25401 

089106 

32 

29 

926119 

24538 

999985 

02 

24540 

078866 

3i 

3o 

940842 

23733 

999983 

02 

940858 

28735 

059142 

3o 

3i 

7.955082 

22980 

9.999982 

02 

7-955100 

22981 

1 2 • 044900 

29 

32 

Q68870 

22273 

999981 

02 

968889 

22275 

o3iiii 

28 

33 

982233 

21608 

999980 

02 

982253 

21610 

017747 

27 

34 

995198 

20981 
20J90 

999979 

02 

995219 

20988 
20892 

004781 

26 

33 

8-007787 

999977 

02 

8-007809 

II.992IQI 
979956 

25 

36 

020021 

1 983 1 

999976 

02 

020044 

19888 

24 

s 

031919 

19302 

999975 

02 

081945 

19805 

968055 

23 

043 00 I 

18801 

999973 

02 

048327 

18808 

956473 

23 

39 

054781 

18325 

999972 

02 

054809 

18827 

945191 

21 

40 

065776 

17872 

999971 

02 

o658o6 

17874 

984194 

20 

4i 

8-076500 

17441 

9-999969 

02 

8-076531 
086997 

17444 

11.928469 
918008 

10 

42 

086963 

17031 

999968 

02 

17084 

18 

43 

097183 

16639 
16265 

999966 

02 

097217 

16642 

002783 

\l 

44 

10716-7 
116926 

999964 

o3 

107203 

16268 

802797 
888087 

45 

15908 

999968 

o3 

116968 
126310 

15910 

i5 

46 

1 2647 1 

15366 

999961 

08 

15568 

878490 

14 

47 

i358io 

15238 

999950 

03 

i8585i 

i524i 

864149 

i3 

48 

144953 

14924 

999958 

o3 

144996 
158932 

14927 

855oo4 

12 

49 

153907 

14622 

999956 

o3 

14627 

846048 

11 

5o 

162681 

14333 

999954 

o3 

162727 

14886 

887273 

10 

5i 

8-171280 

i4o54 

9.999952 

o3 

8-171828 

i4o57 

11.828672 

9 

52 

179713 

13786 

999950 

o3 

188086 

13790 

820287 

a 

53 

187983 

i3529 

999948 

o3 

18382 

811964 
808844 

I 

54 

196102 

18280 

999946 

o3 

196156 

18284 

53 

204070 

18041 

999944 

o3 

204126 

18044 

795874 

5 

56 

2ii8o5 

1 28 10 

999942 

04 

211953 

12814 

788047 

4 

57 

219581 

12587 

999940 

04 

219641 

12590 

780859 

3 

58 

227134 

12872 

999988 

04 

227195 

12876 

772805 

a 

59 

234557 

12164 

999936 

04 

284621 

12168 

765379 

I 

6o 

241855 

11963 

999934 

04 

24I92I 

11967 

758079 

0 
1 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

89" 

Ta 

BLE  IL 

LOGARITHMIC 

SINES,  TANGENTS,  ETC. 

1» 

1 

1 

Sine. 

1   »• 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

8-241855 

1 1 963 

9-999934 

04 

8-241921 

11967 

11-758079 
730898 
743835 

60 

I 

249033 

1 1768 

999932 

04 

249102 

11772 

% 

2 

236094 

11D80 

999929 

04 

256i65 

11384 

3 

263o42 

1 1 398 

99992] 

04 

263ii5 

1 1402 

736885 

57 

4 

.  269881 

11221 

9999" 

04 

269956 

11225 

730044 

56 

5 

276614 

iio5o 

999922 

04 

276091 

iio54 

723309 

55 

6 

283243 

10883 

999920 

04 

283323 

10887 

716677 

54 

7 

289773 

10721 

999918 

04 

289856 

10726 

710144 

53 

8 

296207 

io565 

999913 

04 

296292 
302634 

10570 

703708 
697366 

52 

9 

302546 

io4i3 

999913 

04 

10418 

5i 

10 

308794 

10266 

999910 

04 

308884 

10270 

691116 

5o 

II 

8-314954 

10122 

9.999907 

04 

8-3i5o46 

10126 

11-684954 

% 

12 

321027 

9982 

999903 

04 

321122 

9087 

678878 

i3 

327016 

9847 

999902 

04 

327114 

9831 

672886 

47 

14 

332924 

97'4 

999899 

o5 

333025 

97'9 

666975 

46 

i5 

338753 

9386 

999897 

03 

338856 

9590 

661144 

45 

i6 

344504 

9460 

999894 

o5 

344610 

9465 

655390 

44 

17 
i8 

35oi8i 
355783 

9338 
9219 
9103 

999891 
999888 

o5 
o5 

350289 
355893 
36 1430 

9343 
9224 

649711 
644105 

43 

42 

>9 

36i3i5 

999885 

o5 

9108 
6995 

638570 

41 

20 

366777 

8990 

999882 

o5 

366895 

633io5 

40 

21 

8-372171 

8880 

9-999879 

o5 

8-372292 

8885 

11-627708 

39 

22 

377499 

8772 

999876 

o5 

377622 

8777 

622378 

38 

23 

3^2762 

8667 

999873 

o5 

382889 

8672 

617111 

37 

24 

387962 

8d64 

999870 

o5 

388092 

8570 

611908 

36 

2') 

393101 

8464 

999867 

o5 

393234 

8470 

606766 

35 

26 

398179 

8366 

999864 

o5 

398315 

8371 

601685 

34 

27 

4o3i99 

8271 

999861 

o5 

4o3338 

8276 

596662 

33 

28 

408161 

8177 

999858 

o5 

4o83o4 

8182 

591696 

32 

29 

4i3o68 

8086 

999854 

o5 

4i32i3 

8091 

586787 

3i 

3o 

4i79'9 

7996 

999851 

06 

418068 

8002 

581932 

3o 

3i 

8-422717 

79°9 
7823 

9-999848 

06 

8-422860 

7914 

ii-377i3i 

29 

32 

427462 

999844 

06 

427618 

783o 

572382 

28 

33 

432 1 56 

7740 

999841 

06 

4323 1 5 

7743 

567685 

27 

34 

436800 

7657 

999838 

06 

436962 

7663 

563o38 

26 

35 

441394 

7577 

999834 

06 

441360 

7383 

558440 

25 

36 

445941 

7499 

999831 

06 

446110 

75o5 

553890 

24 

^7 

45o44o 

7422 

999827 

06 

45o6i3 

7428 

549387 

23 

38 

454893 

7346 

999824 

06 

455070 
459481 
463849 

7352 

544930 

22 

39 

459301 

7273 

999820 

06 

7279 

540319 

21 

40 

.  463665 

7200 

999816 

06 

7206 

536i5i 

ao 

4i 

8.467985 

7129 

9-999813 

06 

8-468172 

7135 

11-531828 

"9 

42 

472263 

7060 

999809 
999805 

06 

472454 

7066 

527546 

18 

43 

476493 

6991 

06 

476693 

6998 
6931 

523307 
519108 

17 

44 

480693 

6924 

999801 

06 

48089a 

16 

45 

484848 

6859 

999797 

07 

483030 

6865 

514930 

i5 

4b 

488963 

6704 
6731 

999794 

07 

489170 

6801 

5io83o 

14 

47 

493040 

999790 
999786 

07 

493230 

6738 

506750 

i3 

48 

497078 

6669 

07 

497293 

6676 

501707 

12 

49 

5oio8o 

6608 

999782 

07 

501298 

66i5 

49870a 

11 

5o 

5o5o45 

6548 

999778 

07 

503267 

6555 

494733 

10 

5i 

8-508974 

6489 

9-999774 

07 

8-509200 
5 13098 

64o6 
64J9 

11-490800 

I 

52 

512867 

6431 

999769 

07 

486902 

53 

516726 

6375 

999763 

07 

516961 

6382 

483o39 

1 

54 

52o55i 

63 19 

999761 

07 

520790 
524586 

6326 

479210 

6 

55 

524343 

6264 

999757 

07 

6272 

473414 

5 

56 

528102 

6211 

999753 

07 

528349 

6218 

47i65i 

4 

57 

531828 

6i58 

999748 

07 

532080 

6i65 

467920 

3 

58 

535523 

6106 

999744 

07 

535779 

6ii3 

464221 

a 

59 

539186 

6o55 

999740 

07 

539447 
54J084 

6062 

460553 

1 

60 

542819 

6004 

999735 

07 

601  a 

436916 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang.    j  '  1 

"91^ 

,' 

>8°  i 

20 
2° 

LOviARITHxMIC  SINES, 

TANGENTS,  ETC 

Tabie 

71 

' 

177*  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

f 

0 

8-542819 

G004 

9-999735 

07 

8 .543084 

6012 

11-456916 
453309 

60 

I 

546422 

5955 

999731 

07 

546691 

5962 

59 

2 

549995 

5906 

999726 

°1 

550268 

5914 

449732 

58 

3 

553539 

5858 

999722 

08 

553817 

5866 

446 1 83 

57 

4 

557054 

58ii 

999717 

08 

557336 

5819 

442664 

56 

5 

56o54o 

5765 

999713 

08 

560828 

5773 

439172 

55 

6 

563999 

5719 

999708 

o8 

564291 

5727 

435709 

54 

I 

567431 

5674 

999704 

08 

567727 

5682 

432273 

53 

570836 

563o 

999699 

08 

571137 

5638 

428863 

52 

9 

574214 

5587 

999694 

08 

574520 

5595 

425480 

5i 

10 

577566 

5544 

999689 

08 

577877 

5532 

422123 

5o 

11 

8-580892 

55o2 

9-999685 

08 

8.581208 

55io 

11-418792 

49 

48 

12 

584193 

5460 

999680 

08 

584514 

5468 

415486 

i3 

587469 

5419 

999675 

08 

587795 

5427 

4i22o5 

47 

i4 

590721 

5379 

999670 

08 

591031 

5387 

408949 

46 

i5 

593948 

5339 

999665 

08 

594283 

5347 

405717 

45 

i6 

597 1  52 

53oo 

999660 

08 

597492 

53  08 

4o25o8 

44 

17 

6oo332 

5261 

999655 

08 

600677 

5270 

399323 

43 

i8 

603489 
606623 

5223 

999650 

08 

6o3839 

5232 

396161 

42 

'9 

5i86 

999645 

09 

606978 

5104 
5i58 

3q3o22 

4! 

20 

609734 

5i4g 

999640 

09 

610094 

309906 

40 

21 

8-612823 

5lI2 

9-999635 

09 

8-6i3i89 

5l2l 

11-386811 

39 

22 

615891 

5076 

999629 

09 

616262 

5o85 

383738 

38 

23 

618937 

5o4i 

999624 

09 

619313 

5o5o 

380687 

37 

24 

621962 

5oo6 

999619 

09 

622343 

5oi5 

377657 

36 

25 

624965 

4972 

999614 

09 

623352 

49S1 

374648 

35 

26 

627948 

4938 

999608 

09 

628340 

4947 

371660 

34 

3 

630911 

4904 

999603 

09 

63i3o8 

4913 

368692 

33 

633854 

4871 

999597 

09 

634256 

4.syo 

365744 

32 

2g 

636776 

4839 

999392 

09 

637184 

4848 

362816 

3i 

3o 

639680 

4806 

999386 

09 

640093 

4816 

359907 

3o 

3i 

8-642563 

4775 

9-999581 

09 

8-642982 

47B4 

11-357018 

29 

32 

645428 

4743 

999375 

09 

645853 

4753 

354147 

28 

33 

648274 

4712 

999370 

09 

648704 

4722 

351296 

27 

34 

65iio2 

4682 

999564 

09 

631537 

4691 

348463 

26 

35 

6539.1 

4652 

999338 

10 

654352 

4661 

345648 

25 

36 

606702 

4622 

999553 

10 

657149 

463 1 

342851 

24 

37 

659475 

4592 

999547 

10 

659928 

4602 

340072 

23 

38 

662230 

4563 

999?4i 

10 

662689 

4573 

337311 

22 

39 

664968 

4535 

999335 

10 

665433 

4544 

334367 

21 

4o 

667689 

45o6 

999529 

10 

668160 

4526 

331840 

20 

4i 

8-67o3o3 
673080 

4479 

9-999524 

10 

8-670870 

4488 

II. 329130 

'9 

42 

445 1 

999318 

10 

673563 

4461 

326437 

iS 

43 

675751 

4424 

999312 

10 

676239 

4434 

323761 

17 

44 

678405 

4397 

999306 

10 

678900 

4417 

321100 

16 

45 

681043 

4370 

999300 

10 

681344 

4380 

3 18456 

i5 

46 

683665 

4344 

999493 

10 

684172 

4354 

3 15828 

14 

47 

686272 

43i8 

999487 

10 

686784 

4328 

3i32i6 

i3 

48 

688863 

42Q2 

999481 

10 

689381 

43o3 

310619 

12 

49 

691438 

4267 

999475 

10 

691963 

4277 

3o8o37 

11 

DO 

693998 

4242 

999469 

10 

694329 

4232 

305471 

10 

5i 

8.696543 

4217 

9-999463 

11 

8-697081 

4228 

11.302919 

3oo383 

9 

52 

699073 

4192 

999456 

11 

699617 

42o3 

8 

53 

701589 

4168 

999430 

11 

702139 

4170 

297861 

7 

54 

704090 

4144 

999443 

u 

704646 

4i55 

295354 

6 

55 

706577 

4121 

999437 

11 

707140 

4i32 

292860 

5 

56 

709049 

4097 

999431 

u 

709618 

4108 

290382 

4 

3 

7ii5o7 

4074 

999424 

li 

712083 

4o85 

287917 

3 

713952 

4o5i 

999418 

II 

714534 

4062 

285466 

a 

59 

716383 

4029 

999411 

II 

716972 

4040 

283028 

I 

60 

718803 

4006 

999404 

II 

719396 

4017 

280604 

0 

'  1   Cosine. 

D. 

Sine. 

D. 

Colang. 

D. 

Tang. 

' 

92° 

570 

Table  11.    LOGARITHMIC  SINEfe 

,  TANGENTS,  ETC. 

rl 

3° 

/ 

0 

176* 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang.     ' 

8.718800 

4006 

9-999404 

11 

3.719396 

4017 

1 1  -  280604 

60 

I 

721204 

3984 

999398 

II 

721806 

3993 

278194 

% 

2 

723595 

3g62 

999391 

II 

724204 

3974 

273796 

3 

725972 

3941 

999384 

11 

726588 

3952 

273412 

57 

4 

728337 

3919 

999378 

II 

728959 

3930 

271041 

56 

5 

730688 

3ir'98 

999371 

11 

731J17 

3909 

268683 

55 

6 

733027 

3877 

999364 

12 

733663 

3889 

266337 

54 

7 

735354 

3857 

999357 

12 

735096 
7383n 
740626 

3868 

264004 

53 

8 

737667 

3836 

999350 

12 

3848 

261683 

5a 

9 

739969 

38i6 

999343 

12 

3827 

259374 

5i 

10 

742259 

3796 

999336 

12 

742922 

3807 

257078 

5o 

II 

8-744536 

^^I^ 

9-999329 

12 

3:745207 

3787 

11-254793 

% 

12 

746802 

3756 

999322 

12 

747479 

3768 

232321 

i3 

749055 

3737 

9993 1 5 

12 

749740 

3749 

230260 

47 

14 

751297 

^'2 

999308 

12 

751989 

3729 

248011 

46 

i5 

75352S 

3698 

999301 

12 

734227 

3710 

243773 

45 

i6 

755747 

3679 

999204 

12 

756453 

3692 

243347 

44 

17 

757955 

3661 

999287 

12 

758668 

3673 

241332 

43 

i8 

7601 5 1 

3642 

999279 

12 

760872 

3655- 

239128 

42 

19 

762337 

3624 

999272 

12 

763o65 

3636 

236935 

41 

20 

764511 

36o6 

999263 

12 

765246 

36i8 

234754 

40 

21 

8.766675 

3588 

9-999257 

12 

8-767417 

36oo 

11-232583 

39 

22 

768828 

3570 

999250 

i3 

769578 

3583 

23o422 

38 

23 

770970 

3553 

999242 

i3 

771727 

3565 

228273 

37 

24 

773101 

3535 

999235 

i3 

773866 

3548 

;226i34 

36 

25 

775223 

35i8 

999227 

i3 

773995 

353 1 

224005 

35 

26 

777333 

35oi 

999220 

i3 

778114 

35i4 

221886 

34 

27 

779434 

3484 

9992 1 2 

i3 

780222 

3497 

219778 

33 

28 

781524 

3467 

999205 

i3 

782320 

3480 

217680 

32 

29 

783605  . 

345i 

999197 

i3 

784408 

3464 

215592 

3i 

3o 

785675 

343 1 

999189 

i3 

786486 

3447 

2i35i4 

3o 

3i 

8.787736 

34i8 

9-999181 

i3 

8-788554 

343 1 

1 1  -  2 1 1 446 

29 

32 

789787 
791828 

3402 

999174 

i3 

790613 

3414 

209387 

28 

33 

3386 

999166 

i3 

792662 

3399 

207338 

27 

34 

793859 

^370 

999158 

i3 

794701 

3383 

203299 

26 

35 

795881 

3354 

999 1 5o 

i3 

796731 

3368 

203269 
201248 

25 

36 

797894 

3339 
3323 

999 '42 

i3 

798752 

3352 

24 

37 

799897 

999134 

i3 

800763 

3337 

199237 

23 

38 

801892 

33o8 

999126 

i3 

802765 

3322 

197235 

22 

39- 

803876 

3293 

999118 

i3 

804758 

3307 

193242 

21 

40 

8o5852 

3278 

9991 10 

i3 

806742 

3292 

193238 

20 

41 

8-807819 

3263 

9-999102 

i3 

8-808717 

3278 

11-191283 
189317 

;? 

42 

809777 

3249 

9990^4 
999086 

14 

8 10683 

3262 

43 

811726 

3234 

14 

812641 

3248 

187359 

17 

44 

813667 

32'9 

999077 

14 

814589 

3233 

18541 1 

16 

4) 

815599 

3203 

999069 

14 

816529 

32ig 

183471 

i5 

46 

817522 

3191 

999061 

14 

818461 

3203 

181539 

14 

47 

819436 

3177 

999053 

14 

820384 

3191 

179616 

i3 

48 

821343 

3i63 

999044 

14 

822298 

3.77 

177702 

la 

49 

823240 

3i49 

999o36 

14 

824203 

3i63 

173795 

11 

DO 

825i3o 

3i35 

999027 

14 

826103 

3i5o 

173897 

10 

31 

8-827011 

3l22 

9-999019 

14 

8-827992 
829874 

3i36 

1 1. 1 72008 

I 

32 

828884 

3ioS 

999010 

14 

3i23 

170126 

53 

830749 

3095 

999002 

14 

831748 

3iio 

168252 

7 

54 

832607 

3o82 

■  998993 

14 

8336i3 

3096 

166387 

6 

55 

834456 

3069 

99S9S4 

14 

835471 

3o83 

164529 

5 

56 

836297 
838 i3o 

3o56 

9<)S976 

i4 

837321 

3070 

162679 

4 

37 

3o43 

99^967 

i5 

839163 

3o57 

160837 

3 

58 

839q56 

3o3o 

99S958 

i5 

840098 

3o45 

159002 

a 

59 

841774 

3017 

998930 

i5 

842825 

3o32 

137175 

I 

60 

843585 

3oco 

998941 

i5 

844644 

3019 

135356 

0 

r 

Coeine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

t 

ItS^ 

) 

Bft" 

22 
4» 

LOGARITHMIC  SINES, 

TANGENTS,  ETC 

Table  II. 

IIS" 

1 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

8-843585 

3oo5 

9.998941 

i5 

8-844644 

8019 

11 -155356 

60 

I 

845387 

2992 

998982 

i5 

846455 

3007 

153545 

59 

2 

847183 

2980 

998923 

i5 

848260 

2995 

i5i74o 

58 

3 

848971 

2967 

998914 

i5 

85oo57 

2982 

149943 
148154 

57 

4 

85o75i 

2955 

998905 
998896 

i5 

85 1846 

2970 

56 

5 

852523 

2943 

i5 

853628 

2958 

146872 

55 

6 

854291 

2981 

998887 

i5 

8554o3 

2946 

144597 

54 

I 

856o49 

2919 

998878 

i5 

857171 

2935 

142829 

53 

857801 

2907 

008869 

i5 

858932 

2928 

141068 

52 

9 

859546 

2896 

998860 

i5 

860686 

2911 

189814 

5i 

10 

861283 

2884 

998851 

i5 

862433 

2900 

187567 

5o 

II 

8-863074 

2873 

9 '998841 

i5 

8-864173 

2888 

11 -135827 

49 

12 

864738 

2861 

998832 

i5 

865906 

2877 

184094 

48 

i3 

866455 

285o 

998823 

16 

867682 

2866 

182868 

47 

i4 

868 1 65 

2839 

998813 

16 

869851 

2854 

180649 

46 

i5 

869868 

2828 

998804 

16 

871064 

2843 

1 28936 

45 

i6 

871565 

2817 

998795 

16 

872770 

2882 

127280 

44 

17 

873255 

2806 

998785 

16 

874469 

2821 

125531 

43 

i8 

874938 

2786 

998776 

16 

876162 

2811 

128838 

42 

19 

876615 

998766 

16 

877849 

2800 

122l5l 

41 

20 

878285 

2773 

998757 

16 

879529 

2789 

1 2047 1 

40 

21 

8-879949 

2763 

9-998747 

16 

8-881202 

2779 

11-118798 
117181 

39 

22 

881607 

2752 

998738 

16 

882869 

2768 

38 

23 

883258 

2742 

998728 

16 

884530 

2758 

1 1 5470 

37 

24 

884903 

2731 

998718 

16 

886185 

2747 

Ii38i5 

36 

25 

886542 

2721 

998708 

16 

887888 

2787 

112167 

35 

26 

888174 

2711 

998699 

16 

889476 

2727 

iio524 

34 

S 

889801 

2700 

998689 

16 

891112 

2717 

108888 

33 

891421 

2690 

998679 

16 

892742 

2707 

107258 

32 

29 

893035 

2680 

998669 

17 

894866 

2607 
2687 

io5634 

3i 

3o 

894643 

2670 

998659 

17 

895984 

104016 

3o 

3i 

8-896246 

2660 

9  -  998649 

17 

8-897596 

2677 

II  102404 

so 

32 

897842 

265i 

998689 

17 

899208 

2667 

100797 

28 

33 

899432 

2641 

998629 

17 

900808 

2658 

099197 

27 

34 

901017 

263 1 

998619 

•7 

902808 
908987 

2648 

097602 

26 

35 

902596 

2622 

998609 

17 

2638 

096018 

25 

36 

904169 

2612 

998599 

17 

9o5d7o 

2629 

094480 

24 

S 

905736 

26o3 

998589 
998578 

17 

907147 

2620 

092853 

23 

907297 

2593 

17 

908719 
910285 

2610 

091 281 

23 

39 

908853 

2584 

998568 

17 

2601 

089715 

.21 

40 

910404 

2575 

998558 

17 

911846 

2592 

088 1 54 

20 

41 

8-911949 

2566 

9-998548 

17 

8-913401 

2583 

I I • 086599 

10 

18 

42 

913488 

2556 

998537 

17 

914951 

2574 

o85o49 

43 

9l5022 

2547 

998527 

17 

916495 

2565 

o835o5 

17 

44 

9i655o 

2538 

998516 

18 

918084 

2556 

081966 

16 

45 

918073 

2529 

9g85o6 

18 

919568 

2547 

080482 

i5 

46 

919591 

2520 

998495 

18 

921096 

2538 

078904 
077881 

14 

47 

921103 

25l2 

998485 

18 

922619 

258o 

i3 

48 

922610 

25o3 

998474 

18 

924i36 

2521 

075864 

12 

49 

924112 

2494 

998464 

18 

925649 

25l2 

074351 

11 

5o 

925609 

2486 

998453 

18 

927156 

25o3 

072844 

10 

5i 

8-927100 

2477 

9-998442 

18 

8-928658 

24o5 

11.071842 

I 

52 

928587 

2469 

998481 

18 

980*55 

2486 

069845 

53 

980063 

2460 

998421 

18 

981647 

2478 

068853 

1 

54 

93 1 544 

2452 

998410 

18 

988134 

2470 

066866 

6 

55 

933oi5 

2443 

t^% 

18 

984616 

2461 

065884 

5 

56 

934481 

2435 

18 

986093 

2453 

o6390'2 

4 

57 
58 

935942 
937398 

2427 
2419 

9?836Z 

18 
18 

987565 
989082 

2445 
2437 

062433 
060968 

3 
a 

59 

9388DO 

2411 

998355 

18 

940494 
941952 

2480 

059306 

I 

60 

940296 

2403 

998344 

18 

2421 

o58o48 

0 

1 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

9-t« 

> 

86«  1 

Table  II.    LOGARITHMIC  SINES 

,   TANGENTS,  ETC. 

2z\ 

5° 

1740  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Colang. 

1 

0 

8-940296 

24o3 

9-998344 

'9 

8-941952 

2421 

11 -058048 

60 

I 

941733 

2394 

998333 

>9 

943404 

24i3 

056596 

?? 

2 

943174 

23a7 

998322 

19 

944852 

24o5 

o55i43 

3 

944606 

2379 

9983 1 1 

19 

946295 
947734 

2397 

o537o5 

57 

4 

946034 

2371 

998300 

•9 

2390 

052266 

56 

5 

947456 

2363 

998289 

19 

949168 

2382 

o5o832 

55 

6 

948874 

2355 

998277 

19 

950597 

2374 

049403 

54 

7 

950287 

2348 

998266 

19 

952021 

2366 

047979 

53 

8 

931696 

4340 

998255 

19 

953441 

236o 

046359 

52 

9 

953100 

2332 

998243 

"9 

954856 

235i 

045 I 44 

5i 

10 

954499 

2325 

998232 

"9 

956267 

2344 

043733 

5o 

II 

8-9558o4 

23i7 

9-998220 

19 

8-957674 

2337 

11-042326 

% 

12 

957284 

23lO 

998209 

'9 

959075 

2329 

2323 

040925 

i3 

938670 

2302 

998197 
998186  . 

19 

960473 

039627 
o38i34 

47 

i4 

960052 

2295 

19 

961866 

23i4 

46 

i5 

961429 

2288 

998174 

19 

963255 

23o7 

036745 

45 

i6 

962801 

2280 

998163 

'9 

964639 

23oo 

o3536i 

44 

'I 

964170 

2273 

998151 

19 

966019 

22o3 

033981 

43 

i8 

965534 

2266 

998139 
998128 

20 

967394 

2286 

032606 

42 

19 

966893 

2259 

20 

968766 

2279 

o3i234 

41 

20 

968249 

2252 

9981 16 

20 

970133 

2271 

029867 

40 

21 

8 -969600 

2244 

9-998104 

20 

8-971496 

2265 

ii-0285o4 

39 

22 

970947 

2238 

998092 

20 

972835 

2257 

027145 
023791 

38 

23 

972289 

223l 

998080 

20 

974209 

225l 

37 

24 

973628 

2224 

998068 

20 

975560 

2244 

024440 

36 

25 

974962 

2217 

998056 

20 

976906 

2237 

023094 

35 

26 

976293 

2210 

998044 

20 

978248 

223o 

021762 

34 

11 

977619 

2203 

998032 

20 

979586 

2223 

020414 

33 

978041 

2197 

998020 

20 

980921 

2217 

019079 

32 

29 

980259 
981573 

2190 

998008 

20 

982251 

2210 

017749 
016423 

3i 

3o 

2183 

997996 

20 

983577 

2204 

3o 

3i 

8-982883 

2177 

9-997984 

20 

8-984899 

2197 

ii-oi5ioi 

29 

32 

984189 

2170 

997972 

20 

986217 

2191 

013783 

28 

33 

983491 

986789 
988083 

2i63 

997959 

20 

987532 
988842 

2184 

012463 

27 

34 

2i57 

997947 

20 

2173 

oiii58 

26 

35 

2i5o 

997933 

21 

990149 

2171 

OOnSSl 

23 

36 

989374 

2144 

997922 

21 

991451 

2i65 

008349 

24 

u 

990660 

2i33 

997910 

997697 
997885 

21 

.992730 

2138 

007250 

23 

991943 

2l3l 

21 

994045 

2132 

003955 

22 

39 

993222 

2125 

21 

995337 

2146 

004663 

21 

40 

994497 

2II9 

997872 

21 

996624 

2140 

003376 

20 

41 

8-995768 

2II2 

9-997860 

21 

8-997908 

2i34 

11-002092 

19 
18 

42 

997036 

2106 

997847 

21 

999188 

2127 

000812 

43 

998299 

2100 

997833 

21 

9-000465 

2I2I 

10 -990535 
998262 

17 

44 

999560 

20p4 

997822 

21 

001738 

2Il5 

16 

45 

9-000816 

2087 

997809 

21 

oo3oo7 

2100 

2io3 

996993 

i5 

46 

002060 

2082 

997797 
997784 

21 

004272 

995728 

14 

a 

oo33i8 

2076 

21 

005534 

2097 

994466 

i3 

004563 

2070 

997771 

21 

006792 

2001 

993208 

12 

49 

oo58o5 

2064 

997753 

21 

008047 

2o85 

991953 

II 

5o 

007044 

2o58 

997745 

21 

009298 

2080 

990702 

10 

5i 

9-008278 

2052 

9.997732 

21 

9-010546 

2074 

10-989454 

I 

52 

009510 

2046 

997719 

21 

011790 

2o63 

988210 

53 

010737 

2040 

997706 

21 

0i3o3i 

2062 

986969 

7 

54 

011962 

2o34 

997603 

22 

014263 

2o56 

985732 

6 

55 

oi3i82 

2029 

997680 

22 

015302 

2o5i 

984498 

5 

56 

014400 

2023 

997667 

22 

016732 

2045 

983268 

4 

U 

0i56i3 

2017 

997654 

22 

017939 
019183 

2040 

982041 

3 

016824 

2012 

997641 

22 

2o33 

980817 

3 

59 

oi8o3i 

2006 

997628 

22 

02o4o3 

2028 

979507 

I 

6o 

019235 

2000 

997614 

22 

021620 

3023 

978380 

0 

' 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

[95= 

84<» 

12 


24 

LOGARITHMIC  SINES, 

TANGENTS,  ETC 

Table 

II. 

6° 

r 

3° 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9-019235 

2000 

9-997614 

22 

9-021620 

2023 

10-978380 

60 

I 

020435 

1995 

997601 

22 

022834 

2017 

977166 

59 

3 

02l632 

1989 

997588 

22 

024044 

201  1 

975956 

58 

3 

022825 

1984 

997374 

22 

025251 

2006 

974749 

57 

4 

024016 

1978 

997361 

22 

026455 

2000 

973543 

56 

5 

025203 

1973 

997547 

22 

027655 

1995 

972345 

55 

6 

026386 

1967 

997334 

23 

028852 

1990 

971148 

54 

I 

027567 

1962 

997320 

23 

o3oo46 

1985 

969954 

53 

028744 

1957 

997307 

23 

o3i237 

1979 

968763 
967575 

52 

9 

029918 

igSi 

997493 

23 

032425 

1974 

5i 

10 

031089 

1947 

997480 

23 

033609 

1969 

966391 

5o 

II 

9-o32257 

1941 

9-997466 

23 

9 -03479 1 

1964 

10-965209 

t 

12 

033421 

1936 

997432 

23 

035969 

1958 

964031 

i3 

0345S2 

1930 

997439 

23 

037144 

1953 

962856 

47 

14 

035741 

1925 

907425 

23 

o383i6 

1948 

961684 

46 

i5 

036896 

1920 

9974U 

23 

039485 

1943 

96051 5 

45 

i6 

o38o48 

1915 

997397 

23 

o4o65i 

1938 

959349 

44 

\l 

039197 

1910 

9973o3 

23 

o4i8i3 

1933 

958187 

43 

o4o342 

1Q05 

1B99 

997369 

23 

042973 

1928 

957027 

42 

'9 

041485 

997335 

23 

044 i3o 

1923 

955870 

41 

20 

042625 

1894 

997341 

23 

045284 

1918 

954716 

40 

21 

9-043762 

1889 

9.997327 

24 

9  -  046434 

I913 

10-953566 

39 

22 

044895 

1884 

99731 3 

24 

047582 

1908 

952418 

38 

23 

046026 

1879 

997299 
997285 

24 

048727 

1903 

951273 

37 

24 

047 1 54 

1875 

24 

049869 

1898 

95oi3i 

36 

25 

04B279 

1870 

997271 

24 

o5ioo8 

1893 

1889 

948992 

35 

26 

049400 

i865 

997257 

24 

o52i44 

947836 

34 

11 

o5o5]9 

i860 

997242 

24 

053277 

1884 

946723 

33 

031635 

i855 

997228 

24 

054407 

1879 

945593 

32 

29 

932749 

i85o 

997214 

24 

o55533 

1874 

944465 

3i 

3o 

033859 

1845 

997199 

24 

056659 

1870 

943341 

3o 

3i 

9-054966 

1841 

9-997185 

24 

9-o5778i 
058900 

1 865 

10-942219 

29 

32 

036071 

1 836 

997170 

24 

1869 

941 100 

28 

33 

057I72 

038271 

i83i 

997156 

24 

060016 

1833 

930984 

27 

34 

1827 

997141 

24 

o6ii3o 

l83l 

938870 

26 

35 

039367 

1822 

997127 

24 

062240 

1846 

937760 

25 

36 

060460 

1817 

997112 

24 

063348 

1842 

936652 

24 

3? 

061 55 1 

i8i3 

997098 

24 

064453 

1 837 

935547 

23 

38 

062639 

1808 

997083 

25 

065556 

1 833 

934444 

22 

39 

063724 
064806 

1804 

997068 

25 

066655 

1828 

933345 

21 

40 

1799 

997053 

25 

067732 

1824 

932248 

20 

4t 

9-065885 

1794 

9-997039 

25 

9-068846 

1819 

10-931154 

19 

42 

066962 

1790 

997024 

25 

069938 

l8l3 

q3oo62 

18 

43 

o68o36 

1786 

997009 

25 

071027 
07211.3 

1810 

928078 

17 

44 

069 1 07 

1781 

996994 

25 

1806 

927887 
926803 

16 

45 

070176 

'777 

996979 

25 

073197 
074278 

1802 

i5 

46" 

071242 

1772 

996964 

25 

1797 

925722 

14 

47 
48 

072306 

1768 

996949 

25 

075356 

i703 

924644 

i3 

073366 

1763 

996934 

25 

076432 

1789 

923568 

12 

f^ 

074424 

I75g 

996919 

25 

0775o5 

i7«4 

922495 

II 

5o 

075480 

1753 

996904 

25 

078576 

1780 

921424 

10 

5i 

g-076533 

nSo 

9-996889 

25 

9-079644 

1776 

io-92o356 

I 

52 

077583 

1746 

996874 

25 

080710 

1772 

910290 

53 

078631 

1742 

996858 

25 

081773 

1767 

918227 

I 

54 

079676 

1738 

996843 

25 

082833 

1763 

917167 

55 

0^0719 

1733 

996828 

25 

083891 

1759 

916109 
9i5o53 

5 

56 

081759 

1720 
1725 

996812 

26 

084947 

1755 

4 

ll 

°n?I97 

996707 

26 

086000 

1751 

914000 

3 

083832 

1721 

996782 

26 

087050 

1747 
1743 

912950 

3 

59 

084864 

'7>7 

996766 

26 

088098 

911902 

1 

6o 

080894 

1713 

996751 

26 

089144 

1738 

910856 

0 

1 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

""i)^ 

i 

w 

Table  II.    LOGARITHMIC  SINES 

,  TANGENTS,  ETC. 

ill 

fo 

172° 

/ 

Sine. 

D. 

Cosine. 

D 

Tang. 

n. 

Colang.   1  / 

0 

9 •085894 

1713 

9.996751 

26 

9-089144 

1738 

10-910836 

60 

I 

086922 

1709 

996735 

26 

090187 

1734 

909813  1 

59 

2 

081947 
088970 

1704 

996720 

26 

091228 

1730 

908772 

58 

3 

1700 

996704 

26 

092266 

1727 

907734 

57 

4 

089990 

1696 

996688 

26 

093302 

1722 

906698 

56 

5 

091008 

1692 
1688 

996673 

26 

094336 

1719 

9o5664 

55 

6 

092024 

996657 

26 

095367 

171D 

904633 

54 

I 

093037 

1684 

996641 

26 

096393 

1711 

9o36o5 

53 

094047 

1680 

996625 

26 

097422 

1707 

902378 

52 

9 

O95o56 

1676 

996610 

26 

098446 

1703 

1699 

901554 

5i 

10 

096062 

1673 

,   996594 

26 

099468 

900532 

5o 

II 

9-097065 

1668 

9-996578 

27 

9.100487 

1695 

10-899513 

49 

12 

098066 

1 665 

996562 

27 

ioi5o4 

1691 

898496 

48 

i3 

099065 

1661 

996546 

27 

102519 

16^7 

897481 

47 

U 

100062 

i657 

996530 

27 

103532 

1684 

896468 

46 

i5 

ioio56 

i653 

996514 

27 

104542 

1680 

895458 

45 

i6 

102043 

1649 
1645 

996498 

27 

io555o 

1676 

8o445o 

44 

17 

io3o37 

996482 

27 

106556 

1672 

893444 

43 

18 

104025 

1641 

996465 

27 

107559 
io856o 

i66g 
1 665 

892441 

42 

'9 

loSoio 

i638 

996449 
996433 

27 

891440 

41 

20 

io5g92 

1634 

27 

109559 

1661 

890441 

40 

21 

9-106973 

i63o 

9-996417 

27 

9-110556 

i65S 

10-889444 

39 

22 

107951 

1627 

996400 

27 

iii55i 

1654 

888449 

38 

23 

108927 

1623 

996384 

27 

1 1 2543 

i65o 

887457 

37 

24 

1 0990 1 

1619 

996368 

27 

113533 

i646 

886467 

36 

23 

II 0873 

1616 

996351 

27 

1 14521 

1 643 

885479 
884493 

35 

26 

1 1 1842 

1612 

996335 

27 

11 5507 

1639 

34 

27 

1 12809 

1608 

9963 1 8 

11 

116491 

1636 

883509 

33 

28 

1 13774 

i6o5 

996302 

1 17472 

i632 

882528 

32 

29 

11473-7 
115698 

1601 

996285 

28 

1 1 8452 

1629 

881548 

3i 

3o 

1597 

996269 

28 

119429 

1623 

880571 

3o 

3i 

9-II6656 

1594 

9-996252 

28 

9.120404 

1622 

10-879596 

11 

32 

117613 

1590 
1 5^7 

996235 

28 

121377 

1618 

878623 

33 

118367 

996219 

28 

122348 

i6i5 

877652 

27 

34 

119319 

i583 

996202 

28 

123317 

1611 

876683 

26 

35 

120469 

i58o 

996185 

28 

124284 

1607 

875716 

25 

36 

121417 

1576 

996168 

28 

125249 

1604 

874751 

24 

37 

122362 

1573 

996151 

28 

126211 

1601 

873789 
872828 

23 

38 

i233o6 

1569 

996134 

28 

127172 

i597 

22 

39 

124248 

1 566 

996117 

28 

i28i3o 

1594 

871870 

21 

40 

125187 

1562 

996100 

28 

1 29087 

1591 

870913 

20 

41 

9.I26I25 

1559 

9  -  996083 

29 

9.130041 

1587 

10-869959 

19 

42 

127060 

1 556 

996066 

29 

1 30994 

i584 

869006 

18 

43 

127993 

1 552 

996049 

29 

131944 
132893 

■i58i 

868o56 

17 

44 

128925 
129854 

1549 

996032 

29 

1 577 

867107 

16 

45 

i543 

996015 

29 

133839 

1574 

866161 

i5 

46 

130781 

i542 

995998 

29 

134784 

1571 

865216 

14 

47 

131706 

1539 

993980 

29 

135726 

1567 

864274 

i3 

48 

i3263o 

i535 

995963 

29 

136667 

i564 

863333 

12 

49 

i3355i 

i532 

995946 

29 

137605 

i56i 

862395 

11 

5o 

134470 

i529 

995928 

29 

138542 

i558 

861438 

10 

5i 

9-135387 

i525 

9-995011 

29 

9-139476 

i555 

lo-86o524 

a 

52 

i363o3 

l522 

995894 

29 

140409 

i55i 

859591 

8 

53 

137216 

i5v9 

995876 

29 

i4i34o 

1548 

858660 

7 

54 

i38i28 

i5i6 

995859 

29 

142269 

i545 

857-'3i 

6 

55 

139037 

l5l2 

995841 

29 

143196 

i542 

856^04 

5 

56 

139044 

i5o9 

995823 

29 

144121 

1539 
i535 

855879 

4 

57 
58 

i4o85o 

i5o6 

995806 

29 

145044 

854956 

3 

141754 

i5o3 

995788 

29 

145966 
146885 

i532 

854034 

2 

59 

142655 

i5oo 

993771 

29 

i529 

853113 

1 

60 

143555 

1 496 

995753 

29 

147803 

i526 

832197 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

97^ 

. 

l^ 

26 

LOGARITHMIC  SINES, 

TAXGENTS,  ETC 

Table 

IL 

"8°" 

1 

11^^ 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9-143555 

1496 

9-995753 

3o 

9-147803 

i526 

10-852197 

60 

1 

144453 

1493 

993735 

3o 

148718 

i523 

851282 

% 

2 

145349 

1490 

99^717 

3o 

149632 

1 520 

85o363 

3 

146243 

1487 

993699 

3o 

i5o544 

i5i7 

849456 

57 

4 

147136 

1484 

995681 

3o 

i5i454 

i5i4 

848546 

56 

5 

148026 

1481 

993664 

3o 

152363 

i5ii 

847637 

55 

6 

148015 

1478 

995646 

3o 

153269 

l5o8 

846731 

34 

I 

149802 

1475 

995628 

3o 

154174 

i5c5 

845826 

53 

I 5o686 

1472 

995610 

3o 

153077 

l502 

844923 

5a 

9 

1 5 1 569 

1469 

995591 

3o 

155078 

1499 

8/»4022 

5i 

10 

i5245i 

1466 

993373 

3o 

156877 

1496 

843123 

5o 

II 

9-i5333o 

1463 

9-995555 

3o 

9-157773 

1493 

10-842225 

% 

12 

1 54208 

1460 

995537 

3o 

1 5867 1 

1490 

841329 

i3 

i55o83 

1457 

995519 

3o 

159365 

1487 

840435 

47 

i4 

i55q57 

1454 

995501 

3i 

160457 

1484 

839543 

46 

i5 

i5683o 

i45i 

995482 

3i 

161347 

1481 

838653 

45 

i6 

157700 

1448 

993464 

3i 

162236 

1479 

837764 

44 

17 

1 58569 

1445 

995446 

3i 

i63i23 

1476 

836877 

43 

i8 

139433 

1442 

993427 

3i 

164008 

1473 

833992 

42 

'9 

i6o3oi 

1439 

995409 

3i 

164892 

1470 

835io8 

41 

20 

161164 

1436 

995390 

3i 

165774 

1467 

834226 

40 

21 

9-162025 

1433 

9-995372 

3i 

9-166634 

1464 

10-833346 

39 

22 

162885 

i43o 

995353 

3i 

167532 

1461 

832468 

38 

23 

163743 

1427 

995334 

3i 

16S409 

1458 

83i59i 

37 

24 

1 64600 

1424 

995316 

3i 

169284 

1455 

830716 

36 

23 

165454 

1422 

995297 

3i 

170157 

1453 

820843 

35 

26 

1 663  07 

1419 

993278 

3i 

171029 

i45o 

828971 

34 

27 

167159 

1416 

993260 

3i 

171899 

1447 

82810c 

33 

23 

168008 

I4i3 

99^241 

32 

172767 

1444 

827233 

32 

2q 

168856 

i4io 

993222 

32 

173634 

1442 

826366 

3i 

39 

169702 

1407 

995203 

32 

174499 

1439 

8255oi 

3o 

3i 

9-170547 

i4o5 

9-995184 

32 

9-175362 

1436 

10-824638 

S 

32 

171389 

1402 

993165 

32 

176234 

1433 

823776 

33 

172230 

1399 

995146 

32 

177084 

143 1 

822916 

27 

34 

173070 

1396 

990127 

32 

177942 

1428 

822058 

26 

35 

173908 

1394 

995108 

32 

178799 

1425 

821201 

23 

36 

174744 

1391 

995089 

32 

179653 

1423 

820345 

24 

37 

175578 

1 388 

995070 

32 

i8o5o8 

1420 

819492 

23 

38 

176411 

1 386 

995o5i 

32 

i8i36o 

1417 

818640 

22 

39 

177242 

1 383 

995o32 

32 

182211 

I4i5 

817789 

21 

40 

178072 

i38o 

993013 

32 

i83o59 

1412 

816941 

30 

41 

■9-178900 

1377 

9-994993 

32 

9-183907 

1409 

io-8i6o93 

^2 

42 

179726 

1374 

994974 

32 

184752 

1407 

815248 

iS 

43 

i8o55i 

1372 

994935 

32 

185597 
186439 

1404 

8i44o3 

17 

44 

181374 

1369 

994935 

•32 

1402 

8i356i 

16 

45 

182196 

i366 

994916 

33 

187280 

1399 

812720 

i5 

46 

i83oi6 

1 364 

994896 

33 

188120 

i3o6 

811880 

14 

47 

183834 

i36i 

994877 

33 

18S958 

I3q3 

811042 

i3 

48 

18465 1 

1359 

994857 

33 

189794 

i3oi 

810206 

12 

49 

185466 

i356 

994838 

33 

190629 

1389 

80937: 

11 

5o 

186280 

1353 

994818 

33 

191462 

i386 

8oS538 

10 

5i 

9-187092 

i35i 

9-994798 

33 

9-192294 

1 384 

10-807706 

% 

52 

187903 

i348 

994779 

33 

193124 

i38i 

806876 

53 

188712 

i346 

994759 

33 

193953 

1379 

806047 

I 

54 

18931^ 

i343 

994739 

33 

194780 

1376 

8o5220 

55 

190323 

i34i 

994720 

33 

195606 

1374 

804394 

5 

56 

191130 

i338 

994700 

33 

196430 

1371 

803570 

4 

3 

191933 

1336 

994680 

33 

197253 

1369 

802747 
801926 

3 

192734 

i333 

994660 

33 

19S074 

1 366 

a 

59 

193534 

i33o 

994640 

33 

198894 

i364 

801106 

I 

60 

194332 

i328 

994620 

33 

199713 

i36i 

800287 

0 

1 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

W- 

> 

81° 

Table  II.   LOGARITHMIC  SINES, 

TANGENTS,  ETC. 

271 

^° 

170°  1 

1 

0 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

9-194332 

i32S 

9-994620 

33 

9-199713 

i36i 

10-800287 

60 

I 

195129 

i326 

994600 

33 

200629 

i359 

799471 

5? 

2 

195923 

i323 

994380 

33 

201343 

i356 

798635 

58 

3 

196719 

l32I 

994560 

34 

202169 

i354 

797841 

57 

4 

I97511 

i3i8 

994340 

34 

202971 

i352 

79-020 
796218 

66 

5 

198302 

i3i6 

994319 

34 

203782 

1 349 

65 

6 

199091 

i3i3 

994499 

34 

204692 

'^47 

796408 

54 

7 

199879 

i3ii 

99-^479 

34 

206400 

i345 

794600 

63 

8 

200666 

i3o8 

994450 

34 

206207 

i342 

793793 

62 

9 

20i45i 

i3o6 

994438 

34 

207013 

i34o 

792987 

5i 

10 

202234 

i3o4 

99441 3 

34 

207817 

i338 

792183 

5o 

II 

9-2o3oi7 

i3oi 

9-994398 

34 

9-208619 

i336 

io-79i38i 

49 

12 

203797 

1299 

994377 

34 

209420 

1333 

790680 

48 

i3 

204377 

1296 

994357 

34 

210220 

i33i 

789780 

47 

i4 

2o5354 

1294 

994336 

34 

211018 

1328 

788982 

46 

i5 

2o6i3i 

1292 

994316 

34 

211815 

i326 

788186 

46 

i6 

206906 

1289 

994295 

34 

212611 

i324 

787389 

44 

17 

207679 

1287 

994274 

35 

2i34o5 

l321 

786593 

43 

i8 

208432 

1285 

994234 

35 

214198 

2 1 4989 

i3ig 

786802 

42 

>9 

209222 

1282 

994233 

35 

i3i7 

78601 1 

41 

20 

209992 

1280 

994212 

35 

216780 

i3i5 

784220 

40 

21 

9-210760 

1278 

9-994191 

35 

9-216663 

l3l2 

10-783432 

39 

22 

2Il526 

1275 

994171 

35 

217366 

i3io 

782644 

38 

23 

212291 

1273 

994160 

35 

218142 

i3o8 

781868 

37 

24 

2i3o55 

1271 

994120 

35 

218926 

i3o5 

781074 

36 

25 

2i38i8 

1268 

994108 

35 

219710 

i3o3 

780290 

35 

26 

214579 

1266 

994087 

35 

220492 

i3oi 

779608 

34 

28 

215338 

1264 

994066 

35 

221272 

1299 

778728 

33 

216097 

1261 

994045 

35 

222062 

1297 

777948 

32 

29 

216834 

1239 

994024 

35 

222830 

1294 

777170 

3i 

3o 

217609 

1237 

994003 

35 

223607 

1292 

776393. 

3o 

3i 

9-218363 

1233 

9-993982 

35 

9-224382 

1290 

10-776618 

29 

32 

2191 16 

1253 

993960 

35 

226166 

1288 

774844 

28 

33 

21986S 

125o 

993939 

35 

226929 

1286 

774071 

27 

34 

220618 

1248 

993897 

35 

226700 

1284 

773300 

26 

35 

221367 

1246 

36 

227471 

1281 

772629 

26 

36 

222113 

1244 

99.3875 

36 

228239 

1279 

771761 

24 

37 

222861 

1242 

993854 

36 

229007 

1277 

770993 

23 

38 

2236o6 

1239 

993832 

36 

229773 

1276 

770227 

22 

39 

224349 

1237 

993811 

36 

23o539 

1273 

769461 

21 

40 

225092 

1235 

993789 

36 

23l302 

1271 

76S698 

20 

41 

9-223833 

1233 

9-993768 

36 

9-232065 

1269 

10-767935 

!? 

42 

226573 

123l 

993746 

36 

232826 

1267 

767174 

43 

22731 1 

1228 

993725 

36 

233686 

1266 

766414 

17 

44 

228048 

1226 

993703 

36 

234343 

1262 

765656 

16 

45 

228784 

1224 

993681 

36 

235io3 

1260 

764897 

i5 

46 

229518 

1222 

993660 

36 

236869 

1258 

764141 

14 

47 

230252 

1220 

993638 

36 

2366i4 

1236 

763386 

i3 

48 

2309S4 

1218 

993616 

36 

237368 

1254 

762632 

13 

49 

23i7i5 

1216 

993394 

37 

238120 

1262 

761880 

II 

5o 

232444 

1214 

993572 

37 

238872 

1260 

761128 

10 

5i 

9-233i-2 

1212 

9-993550 

37 

9-239622 

1248 

10-760378 

I 

5? 

233899 

1209 

993528 

37 

240371 

1246 

769629 

53 

234625 

1207 

993506 

37 

24iii3 

1244 

768882 

1 

54 

235349 

1203 

993484 

37 

241866 

1242 

768135 

6 

55 

236073 

I2o3 

993462 

37 

242610 

1240 

767390 

5 

56 

236795 

1201 

993440 

37 

243364 

1238 

7,36646 

4 

57 

2375i5 

1199 

9934 1 S 

37 

244097 

1236 

755903 

3 

58 

233235 

1197 

993396 

^ 

244839 

1234 

755161 

a 

59 

238953 

1193 

993374 

37 

246679 

1232 

754421 

1 

60 

239670 

1.93 

993351 

37 

2463 19 

123o 

753681 

0 

/ 

Cosine. 

D.. 

Sine. 

D. 

Cotang. 

D. 

Tanj. 

^0° 

99"^ 

> 

* 

■  Ill  ■111 

■ "  > 

28 

LOGARITHMIC  SIXES, 

TAXGEXTS,  ETC.   Table  II.  | 

Ti? 

) 

'lC9°  1 

1 

S=ne. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9-239670 

1193 

9-993351 

37 

9-246819 

1280 

10-753681 

60 

1 

240386 

II91 

993329 

37 

247057 

1228 

752943 

59 

3 

34IIOI 

1 189 

993307 

^7 

247794 

1226 

752206 

58 

3 

24I8I4 

"?I 

993284 

f7 

248530 

1224 

751470 

57 

4 

242526 

ii85 

993262 

^ 

249264 

1222 

750786 

56 

5 

843237 

ii83 

993240 

37 

249998 
250780 

1220 

750002 

55 

6 

243947 

1181 

993217 

38 

1218 

749270 

54 

7 

244656 

1 179 

993195 

38 

25i46i 

1217 

748539 

53 

8 

24C363 

"77 

998172 

38 

252191 

12l5 

747809 

52 

9 

246069 

1173 

993149 

38 

232920 

I2l3 

7470.80 

5i 

10 

246773 

1173 

993127 

38 

253648 

I21I 

746332 

5o 

II 

9-247478 

1171 

9-993io4 

38 

9-254874 

1209 

10-743626 

49 

12 

248181 

1169 

998081 

38 

253100 

1207 

744900 

48 

i3 

248883 

1167 

998059 

38 

255824 

1205 

744176 

47 

i4 

249583 

Ii65 

998086 

38 

256547 

i2o3 

743453 

46 

i5 

250282 

ii63 

998013 

38 

257269 

1 201 

742731 

43 

i6 

230980 

1161 

992990 

38 

257990 

I200 

742010 

44 

'2 

251677 

1159 

992967 

38 

258710 

1198 

741290 

43 

i8 

252373 

Ii58 

992944 

.38 

259429 

II96 

740571 

42 

•9 

253067 

ii56 

992921 

38 

260146 

1194 

739854 

41 

20 

■  253761 

Ii54 

992898 

38 

260863 

1 192 

739187 

40 

21 

9-254453 

ll52 

9-992875 

38 

9-261578 

I  loo 
1189 

10-738422 

3o 

22 

255i44 

ii5o 

992852 

38 

262292 

737708 

38 

23 

233834 

1 148 

992829 

39 

263oo5 

1 187 

786995 

37 

24 

256523 

1 146 

992806 

39 

268717 

1185 

786288 

36 

23 

257211 

1 144 

992783 

39 

264428 

1183 

735572 

35 

26 

257898 

1 1 42 

992759 

39 

265i38 

1181 

734862 

34 

27 

258583 

lUi 

992786 

39 

265847 

1170 

734153 

33 

28 

259268 

1 139 

992713 

.39 

266555 

1178 

788445 

32 

29 

259951 

1137 

992690 

39 

267261 

1176 

782789 

3i 

3o 

260633 

ii35 

992666 

39 

267967 

1 174 

782083 

3o 

3i 

g-26i3i4 

ii33 

9-992643 

39 

9-268671 

1172 

10-781329 
730623 

2Q 

32 

261994 

ii3i 

992619 

39 

269875 

1170 
1169 

20 

33 

26:>673 

ii3o 

992396 

39 

270077 

729928 

27 

34 

263351 

1128 

992372 

39 

270779 

1167 

729221 

26 

35 

264027 

1 1 26 

992349 

39 

271479 

ii65 

728521 

25 

36 

264703 

1124 

992523 

39 

272178 

1 164 

727822 

24 

37 

265377 

1122 

992501 

39 

272876 

1162 

727124 

23 

38 

26605 1 

1 1 20 

992478 

40 

273378 

1 160 

726427 

22 

39 

266723 

1119 

992434 

40 

274269 

ii58 

725781 

21 

40 

267395 

1117 

992480 

40 

274964 

1137 

725o36 

20 

41 

9-268065 

Iii5 

9-992406 

40 

9-275658 

1155 

10-724342 

19 

42 

268734 

Iii3 

992882 

40 

276351 

Ii53 

728649 

18 

43 

269402 

nil 

992859 

40 

277043 

ii5i 

722957 

17 

44 

270069 

mo 

992883 

40 

277734 

ii5o 

722266 

16 

45 

270733 

1108 

992811 

40 

278424 

1148 

721576 

i5 

46 

271400 

1 106 

992287 

40 

279118 

1 147 

720887 

14 

47 

272064 

iio5 

992263 

40 

279801 

1 145 

720199 

i3 

48 

272726 

iio3 

992289 

40 

280488 

1 143 

719512 
718826 

12 

49 

273388 

IIOI 

992214 

40 

281174 

II4I 

II 

5o 

274049 

1099 

992190 

40 

281838 

1140 

718142 

10 

5i 

9-274708 

1098 

9-992166 

40 

9-282542 

ii38 

10-717453 

0 

52 

275367 

1096 

992142 

40 

288223 

ii36 

716773 

8 

53 

276025 

1094 

9921 18 

41 

288907 

ii35 

716098 

7 

54 

276681 

1092 

992098 

41 

284388 

1 188 

715412 

6 

55 

277337 

1091 

992069 

41 

285268 

ii3i 

714732 

5 

56 

277991 

io8g 

992044 

41 

285947 

ii3o 

714053 

4 

^7 

278645 

1087 

992020 

41 

286624 

1128 

718876 

3 

58 

279297 

1086 

991996 

41 

287801 

1126 

712699 
712023 

2 

59 

279948 

1084 

991971 

41 

287977 

1125 

I 

6o 

280599 

1082 

991947 

41 

288652 

1123 

711848 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

KK 

)« 

79°  ( 

11< 

3LE  II.    LOGARITHMIC  SINES  TANGENTS,  ETC. 

29  1 

> 

IGS"  1 

/ 

0 

Sine. 

D. 

Cosine. 

1  ^- 

Tan^. 

D. 

Cotang. 

60 

9-2^0099 

1082 

9-991947 

41 

9-288652 

1 1  23 

10-711343 

I 

281248 

io8i 

991922 

41 

289326 

1122 

710674 

59 

2 

2S1B97 

1079 

991897 

41 

289999 

1120 

7 1 000 1 

5il 

3 

282544 

1077 

99 1 873 

41 

29067 1 

1118 

70()329 

57 

4 

283190 

1076 

9,^1848 

41 

•J91342 

1117 

708658 

56 

5 

28J8J6 

1074 

99 1823 

41 

292013 

lll5 

707987 
707318 

55 

6 

284480 

1072 

99' 799 

41 

292682 

1114 

54 

7 

285i24 

1071 

99'774 

42 

293350 

1112 

706630 

53 

8 

280766 

1069 

991749 

42 

294017 

nil 

705983 
7o53i6 

32 

9 

286408 

1067 

991724 

42 

294684 

1109 

5i 

10 

287048 

1066 

991699 

42 

295349 

1 107 

70465 I 

5o 

II 

9-287688 
388326 

1064 

9-991674 

42 

g-296013 

1106 

10-703987 
7o3o23 

40 

12 

io63 

991649 

42 

296677 

1104 

48 

i3 

288964 

1061 

991624 

42 

297339 

iio3 

702661 

47 

14 

289600 

io5q 
io58 

991099 

42 

298001 

1101 

To?^^l 

46 

i5 

290236 

991574 

42 

298662 

1100 

45 

i6 

290870 

io56 

991349 

42 

299322 

1098 

700678 

44 

17 

29i5o4 

io54 

991524 

42 

299980 

1096 

700020 

43 

i8 

292137 

io53 

.  991498 

42 

3oo638 

1095 

699362 

42 

'9- 

292768 

io5i 

991473 

42 

301295 

1093 

698705 

41 

20 

293399 

1030 

991448 

42 

301951 

1092 

698049 

40 

21 

9-294029 

1048 

9-991422 

42 

9.302607 

1000 

10-697393 
696739 

39 
38 

22 

294658 

1046 

991397 

42 

3o326i 

1089 

23 

293286 

1045 

991372 

43 

303914 

1087 

696086 

37 

24 

293913 

1043 

991346 

43 

304567 

1086 

695433 

36 

23 

296539 

1042 

991321 

43 

3o52i8 

1084 

694782 

35 

26 

297164 

1040 

991295 

43 

3o5869 

io83 

694131 

34 

27 

297788 
298412 

1039 

991270 

43 

306319 

1081 

693481 

33 

28 

1037 

991244 

43 

307168 

1080 

692832 

32 

29 

299034 

io36 

991218 

43 

307816 

1078 

692184 

3i 

3o 

299633 

io34 

991193 

43 

3o8463 

1077 

691537 

3o 

3i 

9-300276 

io32 

9-991167 

43 

9.309109 

1075 

10-690891 

29 

32 

300895 

io3i 

991141 

43 

309754 

1074 

690246 

28 

33 

3oi5i4 

1029 

991113 

43 

3 10399 

1073 

689601 

27 

34 

302l32 

1028 

991090 

43 

311042 

1071 

688938 
6883i5 

26 

35 

302748 

1026 

991064 

43 

3ii685 

1070 

23 

36 

3o3364 

1025 

99io38 

43 

312327 

io63 

687673 

24 

37 

3o397g 
304393 

1023 

991012 

43 

312968 

1067 

687032 

23 

38 

1022 

990986 

43 

3i36o8 

io65 

686392 

22 

39 

3o5207 

1020 

990960 

43 

3i4247 

1064 

683733 

21 

40 

3o58i9 

1019 

990934 

44 

3 1 4885 

1062 

683II5 

20 

41 

9 -306430 

1017 

9-990908 

44 

9-313523 

1061 

10-684477 

lo 

42 

307041 

1016 

990882 

44 

3i6i5g 

1060 

683841 

18 

43 

307650 

1014 

990855 

44 

316793 
3 17430 

io58 

683205 

17 

44 

308259 

ioi3 

990829 
990803 

44 

io57 

682570 

yi. 

45 

308867 

lOII 

44 

3 1 8064 

io55 

681936 
68i3o3 

i5 

46 

309474 

lOIO 

.990777 

44 

318697 

io54 

14 

47 

3 10080 

1008 

990750 

44 

3 19330 

io53 

680670 

i3 

48 

3 10685 

1007 

990724 

44 

319961 

io5i 

68oo3g 
679405 

12 

49 

311289 
3 1 1893 

ioo5 

990697 

44 

320392 

io5o 

II 

5o 

1004 

99067  ; 

44 

321222 

1048 

678778 

10 

5i 

9-312495 

ioo3 

9-990643 

44 

9-321831 

1047 

10-678149 

I 

52 

3 1 3097 

lOOI 

91^0618 

44 

322479 

1045 

677521 

53 

313698 

1000 

990391 

44 

323io6 

1044 

676894 

n 

54 

314297 

998 

990365 

44 

323733 

1043 

676267 

6 

55 

314897 

997 

990338 

44 

324358 

io4i 

675642 

5 

56 

3 1 5493 

996 

99031 1 

<5 

324983 

1040 

673017 

4 

57 

316092 
316689 

994 

990485 

45 

323607 

io39 

674393 

3 

5& 

993 

990438 

43 

32623t 

io37 

673769 

3 

59 

317284 

99' 

99043 1 

45 

326853 

io36 

673147 

I 

60 

-3.7879 

990. 

990404 

45 

327475 

1035 

672023 

0 

/ 

Cosine.   | 

D. 

Sine. 

D. 

Cotang.   1 

D. 

Tang. 

1 

"un 

0 

;g<r 

80 

LOGARITHMIC  SINES 

TANGENTS,  ETC.    Table  II. 

12" 

167"  1 

t 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9.317879 

990 

9  990404 

45 

9-327475 

io35 

10.672525 

60 

I 

318473 

988 

990378 

45 

328095 

io33 

671905 

5o 

58 

2 

319066 

9'*7 

99035 1 

45 

328715 

io32 

671285 

3 

319658 

986 

990324 

45 

'  329334 

io3o 

670666 

57 

4 

320249 

9^4 

990297 

45 

329953 

1029 

670047 

56 

5 

320840 

983 

990270 

45 

33o57o 

1028 

669430 

55 

6 

32i43o 

982 

990243 

45 

331187 

1026 

66881 3 

54 

I 

322019 

980 

990215 

45 

33i8o3 

1025 

668197 

53 

322607 

979 

990188 

45 

332418 

1024 

667582 

52 

9 

323I04 

977 

990 1 6 1 

45 

333o33 

1023 

666967 

5i 

10 

323780 

976 

990134 

45 

333646 

I02I 

666354 

00 

II 

9-324366 

975 

9-990107 

46 

9-334259 

1020 

10-665741 

49 

48 

12 

324950 

973 

990079 

46 

334871 

1019 

665120 

i3 

325534 

972 

990032 

46 

335482 

1017 

664518 

47 

i4 

326117 

970 

990025 

46 

336093 

1016 

663907 

46 

i5 

326700 

969 

989997 

46 

336702 

1013 

6632o8 

45 

i6 

327281 

968 

989970 

46 

337311 

I0l3 

662689 

44 

17 

327862 

966 

9H9942 

46 

337(^19 

1012 

662081 

43 

i8 

328442 

965 

989915 

46 

338D27 

lOlI 

661473 

42 

'9 

329021 

964 

989887 

46 

339133 

lOlO 

660867 

41 

20 

329599 

962 

989860 

46 

339739 

1008 

660261 

40 

21 

9-330176 

961 

9-989832 

46 

9-340344 

1007 

10-659656 

39 

22 

330753 

960 

989804 

46 

340948 

1006 

659052 

38 

23 

33 1 329 

958 

989777 

46 

341552 

1004 

658448 

37 

24 

331903 

9?7 

9S9749 

47 

342155 

ioo3 

657845 

36 

23 

332478 

936 

989721 

47 

342757 

1002 

657243 

35 

26 

333o5i 

934 

989693 

47 

343358 

1000 

656642 

34 

27 

333624 

953 

989665 

47 

343958 

999 

656o42 

33 

28 

334195 

952 

989637 

47 

344558 

998 

655442 

32 

29 

334767 

95o 

989610 

47 

345i57 

997 

654843 

3i 

3o 

335337 

949 

989582 

47 

343755 

996 

654243 

3o 

3i 

9-335906 

948 

9-989553 

47 

9-346353 

994 

10-653647 

39 
28 

32 

336475 

946 

989525 

47 

346949 

993 

653o5i 

33 

337043 

945 

989497 

47 

347343 

992 

652455 

27 

34 

337610 

944 

989469 

47 

348141 

991 

65 1 859 

26 

35 

338176 

943 

989441 

47 

3^8735 

990 

65i263 

25 

36 

338742 

941 

989413 

47 

349329 

988 

650671 

24 

37 

339307 

940 

989335 

47 

349922 
35o5i4 

987 

630078 

23 

38 

339871 

939 

989356 

47 

986 

649486 
648894 

22 

39 

340434 

937 

989328 

47 

331 106 

985 

21 

40 

340996 

936 

989300 

47 

351697 

983 

6483o3 

20 

41 

9-3ii558 

935 

9-989271 

47 

9.352287 

982 

10-647713 

19 

42 

342119 

934 

989243 

47 

352876 

98. 

647124 

18 

43 

342679 

932 

989214 

47 

353465 

980 

646535 

«7 

44 

343239 

931 

989186 

47 

354053 

979 

645947 

16 

45 

343797 
344355 

930 

98907 

47 

354640 

977 

645360 

i5 

46 

929 

989128 

48 

355227 

976 

644773 

14 

47 

344gi2 

927 

989100 

48 

3558i3 

973 

644187 

i3 

48 

345469 

926 

989071 

48 

356398 

974 

643602 

12 

49 

346024 

925 

989042 

48 

356982 

973 

643oi8 

11 

DO 

346579 

924 

989014 

48 

337D66 

971 

642434 

10 

5i 

9-347134 

922 

9-988985 

48 

9.358149 

970 

io-64i85i 

I 

52 

347687 

921 

988956 

48 

358731 

969 

641269 

53 

348240 

920 

988927 

48 

359313 

968 

640687 

7 

54 

348792 

919 

988898 

48 

359893 

967 

640107 

6 

55 

349343 

9'7 

988869 

48 

360474 

966 

639526 

5 

56 

349893 

916 

988840 

48 

36io53 

965 

638o47 
638368 

4 

57 

35o443 

9.5 

9S881I 

49 

36i632 

963 

3 

58 

350992 
35i54o 

9'4 

988782 

49 

362210 

962 

637790 

a 

59 

913 

988753 

49 

362787 

961 

637213 

I 

60 

352088 

911 

988724 

49 

363364 

960 

636636 

0 

1 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

102 

0 

770  1 

Table  II.   LOGARITHMIC  SINES 

TANGENTS,  ETC.        31  | 

13° 

160°  1 

/ 

0 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

9-352088 

911 

9-988724 

49 

9-363364 

960 

10-636636 

60 

I 

352635 

910 

988695 

49 

363940 
36451 5 

959 

636o6o 

S 

2 

353i8i 

909 

988666 

49 

958 

635485 

3 

353726 

908 

988636 

49 

365o90 

957 

634910 
634336 

57 

4 

354271 

907 

988607 

988578 

49 

365664 

935 

56 

5 

3548i5 

903 

49 

366237 

934 

633763 

55 

6 

355358 

904 

988543 

49 

366810 

953 

633190 

54 

7 

355qoi 

903 

988519 

49 

367382 

932 

632618 

53 

8 

356443 

902 

988489 

49 

367953 

95i 

632047 

52 

9 

356984 

001 
899 

988460 

49 

368324 

950 

63 1476 

5i 

10 

357324 

988430 

49 

369094 

949 

630906 

5o 

II 

9 .358064 

898 

9-988401 

49 

9  -  369663 

943 

io-63o337 
629768 

49 

12 

3586o3 

897 

988371 

49 

370232 

946 

43 

i3 

359141 

896 

988342 

49 

370799 

945 

629201 

47 

i4 

359678 

893 

988312 

5o 

371367 
371933 

944 

628633 

46 

i5 

36o2i5 

893 

988282 

5o 

943 

628067 

45 

i6 

360752 

892 

988252 

5o 

372499 

942 

627501 

44 

\l 

361287 

891 

988223 

5o 

373064 

941 

626936 

43 

361822 

890 

988193 

5o 

373629 
37419J 

940 

626371 

42 

19 

362356 

889 

988163 

5o 

939 

625807 

41 

20 

362889 

888 

988133 

5o 

374736 

938 

625244 

40 

21 

9-363422 

887 

9-988103 

5o 

9-375319 

937 

10-624681 

39 

22 

363954 

885 

988073 

5o 

375881 

935 

624119 

38 

23 

364485 

884 

988043 

5o 

376442 

934 

623558 

37 

24 

365oi6 

883 

988013 

5o 

377003 

933 

622997 
622437 

621878 

36 

23 

365546 

882 

987983 

5o 

377563 

932 

35 

26 

366075 

881 

987953 

5o 

378122 

931 

34 

S 

366604 

880 

987922 
987892 

5o 

378681 

930 

621319 

33 

367i3i 

879 

5o 

379239 

929 

620761 

32 

29 

367659 
368i85 

877 

987862 

5o 

379797 

928 

620203 

3i 

3o 

876 

987832 

5i 

38o354 

927 

619646 

3o 

3i 

9-368711 

875 

9-987801 

5i 

9-380910 

926 

10-619090 
6i8534 

29 

32 

369236 

874 

9*^777  J 

5i 

381466 

923 

28 

33 

369761 

873 

987740 

5i 

382020 

924 

617980 

27 

34 

•  370285 

872 

987710 

5i 

382573 

923 

617425 

26 

35 

370808 

871 

987679 

5i 

383129 

922 

616871 

25 

36 

37i33o 

870 

987649 

5i 

383682 

921 

6i63i8 

24 

37 

371852 

869 

987618 

5i 

384234 

920 

615766 

23 

38 

372373 

867 

987588 

5i 

384786 

919 

6i52i4 

22 

39 

372894 

866 

987557 
987526 

5i 

385337 

9,8 

614663 

21 

40 

373414 

865 

5i 

385888 

9'7 

614112 

20 

4i 

9-373933 

864 

9-987496 

5i 

9-386438 

9i5 

io-6i3562 

19 
18 

42 

374452 

863 

987465 

5i 

386987 

914 

6i3oi3 

43 

374970 

862 

987434 

5i 

387536 
388084 

9i3 

612464 

17 

44 

375487 

861 

987403 

52 

912 

611916 
6ii369 

16 

4i 

376003 

860 

987372 

52 

388631 

911 

i5 

46 

376519 

859 

987341 

52 

389178 

910 

610822 

14 

47 

377035 

858 

987310 

52 

389724 

900 

610276 

i3 

48 

377549 

837 

987270 
987248 

52 

390270 

908 

609730 

12 

49 

378063 

856 

52 

390815 

907 

609183  ' 

II 

5o 

378577 

854 

987217 

52 

391360 

906 

600640 

10 

5i 

9-379089 

853 

5-987186 

52 

9-391903 

905 

10-608097 

0 

53 

379601 

852 

987155 

52 

392447 

904 

607533 

8 

53 

38oii3 

85i 

987124 

52 

392989 

903 

60701 1 

7 

54 

380624 

85o 

98709a 

52 

393531 

902 

6)6469 

6 

55 

38ii34 

849 

987061 

52 

394073 

901 

605927 
6o5386 

5 

56 

381643 

848 

987030 

52 

394614 

QOO 

4 

^7 

382152 

847 

986998 

52 

395134 

898 

604846 

3 

58 

382661 

846 

986967 
986936 

52 

395694 
396233 

6o43o6 

2 

59 

383 168 

845 

52 

897 

603767 

1 

6o 

383673 

844 

986904 

32 

396771 

896 

603229 

0 

1 

/ 

Cosino. 

D. 

Sine. 

D. 

Cotang. 

D. 

T«ng. 

lOf 

!° 

76»  j 

32 

LOGARITHMIC  SINES 

TANGENTS,  ETC.    Table 

ri 

14° 

166°  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9-383675 

844 

9-986904 

52 

9-396771 

896 

IO-6o3229 

6c 

1 

384182 

843 

9S6873 

53 

397309 

896 

602691 

% 

2 

384687 

842 

986841 

53 

397846 

895 

602134 

3 

385i92 

841 

986809 

53 

398383 

894 

601617 

57 

4 

3856y7 

840 

98677S 

53 

398919 

893 

601081 

56 

5 

386201 

839 

9S6746 

53 

399455 

892 

600545 

55 

6 

386704 

838 

986714 

53 

399990 

891 

600010 

54 

I 

387207 

837 

986683 

53 

400324 

890 

599476 

53 

387709 

836 

986651 

53 

4oio58 

889 

598942 

52 

9 

388210 

835 

986619 

53 

40 1591 

888 

598409 

5i 

10 

3887 II 

834 

986087 

53 

402 1 24 

887 

597876 

5o 

II 

9.3S92II 

833 

9-986555 

53 

9-402656 

886 

iv- 597344 

49 

12 

389711 

832 

986523 

53 

4o3i87 

885 

596813 

48 

i3 

390210 

83 1 

986491 

53 

403718 

884 

596282 

47 

14 

390708 

83o 

986439 

53 

404249 

883 

595731 

46 

i5 

391206 

828 

986427 

53 

404778 

882 

595222 

45 

i6 

391703 

827 

986395 

53 

4o53o8 

881 

594692 

44 

17 

392199 

826 

986363 

54 

4o5836 

880 

594164 

43 

18 

392605 

825 

986331 

54 

4o6364 

879 

593636 

42 

'9 

393191 

824 

986299 

34 

406892 

878 

593108 

41 

20 

393685 

823 

986266 

54 

407419 

877 

592581 

40 

21 

9-394179 

822 

9-986234 

54 

9-407945 

876 

10-592055 

39 

22 

394673 

821 

986202 

54 

408471 

875 

591529 

38 

23 

395166 

820 

986169 

54 

408996 
40932 1 

■874 

591004 

37 

24 

395658 

819 

986137 

54 

S^^ 

590479 
589955 

36 

25 

396150 

818 

986104 

54 

410045 

873 

35 

26 

396641 

817 

986072 

54 

410369 

872 

589431 

34 

27 

3g7l32 

817 

986039 

54 

41 1092 

871 

588908 

33 

28 

391621 

816 

986007 

54 

4ii6i5 

870 

588385 

32 

29 

398III 

8i5 

983974 

54 

412137 

869 

587863 

3i 

3o 

398600 

814 

985942  ■ 

54 

412658 

868 

587342 

3o 

3i 

9-399088 

8i3 

9-985909 

55 

9.413179 

867 

10-586821 

29 

32 

399575 

812 

985876 

55 

413699 

866 

586301 

28 

3i 

400062 

811 

985843 

55 

414219 

865 

585781 

27 

34 

400549 

810 

98581 1 

55 

414738 

864 

585262 

26 

35 

4oio3d 

809 

98577S 

55 

415257 

864 

584743 

25 

36 

40 1 520 

808 

985745 

55 

413775 

863 

584225 

24 

37 

4o20o5 

807 

985712 

55 

416293 

862 

533707 

23 

38 

402489 

806 

985679 

55 

416810 

861 

583190 

22 

39 

402972 

8c5 

983646 

55 

417326 

860 

582674 

21 

40 

403455 

804 

9856i3 

55 

417842 

859 

582158 

20 

41 

9-403938 

8o3 

9-985580 

55 

9-418358 

858 

io-58i642 

10 
18 

42 

404420 

802 

985547 

55 

418873 

857 

581127 

43 

404901 

801 

985514 

55 

419387 

856 

58o6i3 

17 

44 

405382 

800 

985480 

55 

419901 

855 

580099 

16 

45 

405862 

799 

985447 

55 

42o4i5 

855 

579585 

i5 

46 

4ob34i 

7/^ 

985414 

56 

420927 

854 

570073 

14 

tl 

406820 

797 

985381 

56 

421440 

853 

578560 

i3 

.  407299 

796 

985347 

56 

421952 

852 

578048 

12 

49 

407777 

795 

985314 

56 

422463 

85i 

577537 

II 

5o 

408254 

794 

985280 

56 

422974 

85o 

577026 

10 

5i 

9-408731 

794 

9  985247 

56 

9-423484 

849 

io-5765i6 

% 

52 

409207 

793 

985213 

56 

423993 

848 

576007 

53 

409682 

792 

985180 

56 

4245o3 

848 

575407 

7 

54 

410157 

791 

985146 

56 

425oii 

847 

574989 

6 

55 

4io632 

790 

985ii3 

56 

425519 

846 

574481 

5 

56 

4iiio6 

789 

985070 

56 

426027 

845 

573973 

4 

57 

411579 

788 

985045 

56 

426534 

844 

573466 

3 

58 

412032 

787 

985011 

56 

427041 

843 

572959 
572453 

2 

59 

412524 

786 

984978 

56 

427547 

843 

I 

60 

412996 

785 

984944 

56 

428032 

842 

571948 

0 

1 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

\   104 

0 

-5° 

Tadle  II.   LOGARITHMIC  SINES 

TANGENTS,  ETC.        88  | 

15° 

1640  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9-412996 

785 

9-984944 

^ 

9-428052 

842 

10-571948 

60 

I 

413467 

784 

984910 

57 

428558 

841 

571442 

59 

58 

2 

41 3938 

783 

984876 

f7 

429062 

840 

570938 

3 

414408 

783 

984842 

57 

429566 

839 

570434 

a 

4 

414878 

782 

984808 

^^ 

430070 

838 

569930 

5 

4 1 5347 
4i58i5 

781 

984774 

57 

430573 

838 

569427 

55 

6 

780 

984740 

57 

431075 

837 

568925 

54 

7 

416283 

779 

984706 

57 

43i577 

836 

568423 

53 

8 

416751 

778 

984672 

57 

432079 

835 

567921 

52 

9 

417217 

777 

984638 

57 

432580 

834 

567420 

5i 

10 

417684 

776 

984603 

57 

433080 

833 

566920 

5o 

II 

g-4}8\5o 

775 

9-9S4569 

^ 

9-433580 

832 

10-566420 

3 

13 

4 1 861 5 

774 

984533 

57 

434080 

832 

565920 

i3 

419079 

773 

984500 

57 

434579 

83 1 

565421 

47 

14 

419544 

773 

984466 

57 

435078 

83o 

564922 

46 

i5 

420007 

772 

984432 

58 

435576 

829 

564424 

45 

i6 

420470 

771 

984397 

58 

436073 

828 

563927 

44 

17 

420933 

770 

984363 

58 

436370 

828 

563430 

43 

i8 

421395 

760 

984328 

58 

437067 

I'l 

562933 

42 

•9 

4218D7 

768 

984294 

58 

437563 

826 

562437 

4i 

20 

4223i8 

767 

984239 

58 

438059 

825 

561941 

40 

21 

9-422778 

767 

9-9S4224 

58 

9-438554 

824 

io-56i446 

39 

22 

423238 

766 

984190 

58 

439048 

823 

560932 

38 

23 

423697 

765 

984155 

58 

439543 

823 

560457 

37 

24 

424i56 

764 

984120 

58 

44oo36 

822 

559964 

36 

25 

424615 

763 

984085 

58 

440529 

821 

559471 

35 

26 

425073 

762 

984030 

58 

441022 

820 

558978 

34 

3 

425530 

761 

984015 

58 

44i5i4 

819 

558486 

33 

425987 

760 

983981 

58 

442006 

819 

557994 

32 

29 

426443 

760 

983946  • 

58 

442497 

818 

557503 

3i 

3o 

426899 

759 

983911 

58 

442988 

817 

557012 

3o 

3i 

9-427354 

758 

9-983875 

58 

9-443479 
443968 

816 

10-556521 

29 

32 

427809 
428263 

757 

983840 

59 

816 

556o32 

28 

33 

756 

9838o5 

59 

444458 

8i5 

555542 

27 

34 

428717 

755 

983770 

59 

444947 

814 

555o53 

26 

35 

429170 

754 

983735 

59 

445435 

8i3 

554565 

25 

36 

429623 

753 

983700 

59 

445923 

812 

554077 

24 

37 
38 

430075 

752 

983664 

59 

44641 1 

812 

553589 

23 

430327 

752 

983629 

59 

446898 

811 

553io2 

22 

39 

430978 

75i 

983594 

59 

447384 

810 

552616 

21 

40 

431429 

730 

983538 

59 

447870 

809 

552i3o 

20 

41 

9-431879 

749 

9-983523 

59 

9-448356 

809 
808 

io-55i644 

10 

42 

432320 

749 

983487 

59 

448841 

55ii59 

18 

43 

43277S 

748 

983452 

59 

449326 

807 

550674 

17 

44 

433226 

747 

983416 

59 

449810 

806 

550190 

16 

45 

433675 

746- 

983381 

29 

450294 

806 

549706 

i5 

46 

434122 

743 

983345 

59 

430777 

8o5 

540223 

14 

47 

434569 

744 

983309 
983273 

59 

451260 

804 

548740 

i3 

48 

435016 

744 

60 

451743 

8o3 

548257 

12 

49 

435462 

743 

983238 

60 

452225 

802 

547775 

II 

5o 

435908 

742 

983202 

60 

452706 

802 

547294 

10 

5r 

9-436353 

741 

9-983166 

60 

9-453187 

453663 

801 

10-546813 

I 

52 

436798 

740 

983 i3o 

60 

800 

546332 

53 

437242 

740 

983094 
983o58 

60 

454148 

799 

545852 

7 

54 

437686 

730 
738 

60 

454628 

79? 

545372 

6 

55 

438129 

983022 

60 

433107 

455586 

798 

544893 

5 

56 

438572 

737 

982986 

60 

797 

544414 

4 

57 

439014 

736 

982950 

60 

456064 

796 

543936 

3 

58 

439456 

736 

982914 

60 

456542 

796 

543458 

3 

59 

439807 
440338 

735 

982878 

60 

437019 

795 

542981 

I 

60 

734 

982842 

60 

457496 

794 

542304 

0 

/ 

lOt 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D- 

Tang. 

/ 

° 

740  1 

34 

LOGARITHMIC  SINES 

,  TANGENTS,  ETC.    Table  IT.  | 

16 

3 

1 

6-^° 

/ 

1 

Sine. 

D. 

Cosine. 

D. 

Tang. 

1   D. 

Cotang. 

0 

9-440338 

734 

9-982842 

60 

9-457496 

794 

10 -542504 

60 

I 

440778 

733 

982805 

60 

457973 

793 

542027 

59 

2 

441218 

732 

982769 

61 

458449 

793 

54i55i 

58 

3 

441658 

73. 

982733 

61 

458935 

792 

541075 

57 

4 

442096 

73 1 

982696 

61 

459400 

791 

540600 

30 

5 

442535 

73o 

982660 

61 

459875 

790 

540125 

55 

6 

442973 

729 

982624 

6c 

460349 

790 

539651 

54 

7 

443410 

728 

982587 

61 

460823 

780 

539177 

53 

s 

443847 

1  727 

982551 

61 

461297 

78a 

538703 

52 

9 

444284 

727 

982514 

61 

461770 

788 

538230 

5i 

10 

444720 

726 

982477 

61 

462242 

787 

537758 

5o 

II 

9-445i55 

725 

9-982441 

61 

9-462715 

786 

10-537285 

49 

12 

445590 

724 

982404 

61 

463 186 

785 

536814 

48 

i3 

446025 

7i3 

982367 

61 

463658 

785 

536342 

47 

14 

446459 
446893 

723 

982331 

61 

464128 

784 

535872 

46 

i5 

722 

982294 

6i 

464599 

783 

535401 

45 

i6 

447326 

721 

982257 

61 

465069 

783 

534931 

44 

17 

447739 

720 

982220 

62 

465530 

782 

534461 

43 

i8 

448191 

720 

982183 

62 

466008 

781 

533992 

42 

'9 

448623 

719 

982146 

6a 

466477 

780 

533D23 

41 

20 

449054 

718 

982109 

62 

466945 

780 

533o55 

40 

21 

9-449485 

717 

9-982072 

63 

g-467413 

779 
778 

10-532587 

39 

22 

4499' 5 

716 

982035 

62 

467880 

532120 

38 

23 

45o345 

7,6 

981998 

62 

468347 

778 

53 1 653 

37 

24 

450775 

7i5 

981961 

62 

468814 

777 

53 1186 

36 

25 

45i2o4 

7'4 

981924 

62 

469280 

776 

530720 

35 

26 

45i632 

7>3 

981886 

62 

469746 

775 

530254 

34 

27 

452060 

7i3 

981849 

62 

4702 1 1 

775 

529789 

33 

28 

452488 

712 

981812 

62 

^  470676 

774 

529324 
528839 

32 

29 

452915 

711 

981774 

62 

*  47H41 

773 

3i 

3o 

453342 

7,10 

981737 

62 

471605 

773 

52839D 

3o 

3i 

9-453768 

710 

9-981700 

63 

9-472069 

772 

10-527931 

29 

32 

454194 

709 

981662 

63 

472532 

771 

527468 

28 

33 

454619 

708 

981625 

63 

472995 

771 

527005 

27 

34 

455o44 

707 

981587 

63 

473457 

770 

526543 

26 

35 

455469 

707 

981549 

63 

473919 

769 

526081 

23 

36 

455893 

706 

98012 

63 

474381 

769 

525619 

24 

37 

4563 1 6 

7o5 

981474 

63 

474842 

768 

525 1 58 

23 

38 

456739 

704 

981436 

63 

4753o3 

767 

524697 

22 

39 

457162 

704 

981399 

63 

475763 

767 

524237 

21 

40 

457584 

703 

981361 

63 

476223 

766 

523777 

20 

41 

9-458006 

702 

9-981323 

63 

9-476683 

765 

!0-5233i7 

'9 

42 

458427 

701 

981285 

63 

477142 

765 

522858 

18 

43 

458848 

701 

981247 

63 

477601 

764 

522399 

17 

44 

459268 

700 

981209 

63 

478059 

763 

521941 

16 

45 

459688 

699 
698 

98 II 7 1 

63 

478517 

763 

521483 

1 5 

46 

460108 

981 i33 

64 

478975 

762 

521025 

14 

47 

460527 

698 

981095 

64 

479432 

761 

520563 

i3 

48 

460946 

697 

981057 

64 

479889 

761 

5201 11 

12 

49 

46i364 

696 

981019 

64 

48o343 

760 

519655 

11 

5o 

461782 

695 

980981 

64 

480801 

739 

519199 

10 

5i 

9-462199 

695 

9-980942 

64 

9-481257 

759 

10-518743 

9 

52 

462616 

694 

980904 

64 

481712 

758 

518288 

8 

53 

463o32 

693 

980866 

64 

482167 

737 

517833 

7 

54 

463448 

693 

980827 

64 

482621 

737 

517379 
516923 

6 

55 

463864 

692 

980789 

64 

483075 

756 

5 

56 

464279 

691 

980750 

64 

483529 

755 

5i647i 

4 

57 

464694 

690 

9807 1 2 

64 

483982 

755 

5i6oi8 

3 

58 

465 I 08 

690 

980673 

64 

484435 

754 

5 1 5565 

3 

59 

465522 

689 
688 

980635 

64 

484887 

753 

5i5ii3 

I 

60 

465935 

980596 

64 

485339 

753 

514661 

0 

1 

Cosine. 

D.  ! 

Sine. 

D. 

Cotang. 

D.   1 

Tang. 

/ 

106 

0 

>■ 

3° 

Table  II.   LOGARITHillC  SINES 

,  TANGENTS,  ETC. 

86 

17<^ 

162°  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

60 

o 

0-465935 

688 

9'98o5'36 

64 

9-485339 

755 

io-5i466i 

I 

466348 

688 

980553 

64 

48S79I 

752 

514209 
313-58 

69 

58 

3 

466761 

687 

980519 

65 

486242 

75. 

3 

467173 

686 

980480 

65 

486693 

75i 

5i33o7 

57 

4 

467585 

685 

980442 

65 

487143 

75o 

5i2857 

56 

5 

467996 

685 

980403 

65 

487593 

749 

512407 

55 

6 

468407 

684 

980364 

65 

488043 

749 
748 

511937 

54 

7 

468817 

683 

980325 

65 

488492 

5ii3o8 

53 

8 

469227 

683 

9B0286 

65 

488941 

747 

5iio59 

32 

9 

469637 

682 

980247 

65 

489390 

747 

5io6io 

5l 

10 

470046 

681 

980208 

65 

489838 

746 

5ioi62 

5o 

II 

9-470455 

680 

9-980169 

65 

9-490286 

746 

10-509714 

8 

12 

470863 

680 

9801 3o 

65 

490733 

745 

509267 

i3 

471271 

679 

980091 
980002 

65 

491180 

744 

508820 

47 

14 

471679 

678 

65 

491627 

744 

508373 

46 

i5 

4720S6 

678 

980012 

65 

492073 

743 

607927 

43 

i6 

472492 

'^n 

979973 

65 

492519 
492965 

743 

607481 

44 

'7 

472898 

676 

979934 

66 

742 

607035 

43 

i8 

473304 

^A 

979895 

66 

493410 

741 

606690 

42 

19 

473710 

675 

979835 

66 

493854 

740 

606146 

41 

20 

474115 

674 

979816 

66 

4J4299 

740 

606701 

40 

21 

9-4745IO 
47492J 

674 

9-979776 

66 

9-494743 

740 

io-5o5257 

It 

22 

673 

979737 

66 

495186 

739 

604814 

23 

475327 

672 

979697 

66 

49563o 

738 

604370 

37 

24 

473730 

672 

979608 

66 

496073 

737 

603927 

36 

23 

476133 

671 

979618 

66 

49651 5 

737 

5o3485 

35 

20 

476536 

670 

979379 

66 

496957 
-97399 

736 

6o3o43 

34 

'1 

476038 
477^40 

669 

979539 

66 

736 

602601 

33 

28 

669 

979499 
979459 

66 

497841 

735 

502 I  69 

32 

29 

477741 

668 

66 

498282 

734 

601718 

3i 

3o 

478142 

667 

979420 

66 

498722 

734 

601278 

3o 

3i 

9-478542 

667 

9-979380 

66 

9-499163 

733 

10-600837 

29 

28 

32 

478942 
479342 

666 

979340 

66 

499603 

733 

600397 

33 

665 

979300 

67 

5ooo42 

732 

499938 

27 

34 

479741 

665 

979260 

67 

500481 

73i 

499019 

26 

35 

480140 

"664 

979220 

67 

500920 

73 1 

499080 

23 

36 

48o53g 

663 

979180 

67 

5oi359 

730 

498641 

24 

37 

480937 
481334 

663 

979140 

67 

501797 

73o 

498203 

23 

3S 

662 

979100 

67 

5o2235 

729 

497763 

22 

39 

481731 

661 

979059 

^J 

502672 

728 

497328 

21 

40 

4S2128 

661 

979019 

67 

5o3 1 09 

728 

496891 

20 

41 

9-482525 

660 

9-978979 

67 

9-5o3546 

727 

10-496454 

19 

42 

482021 

639 

978539 

67 

503982 

727 

4960 1 8 

id 

43 

483316 

659 
658 

978^98 

67 

5o44i8 

726 

493682 

17 

44 

483712 

978858 

67 

5o4854 

725 

490146 

16 

45 

484107 

657 

978817 

6-' 

505289 

7:^5 

494711 

i5 

46 

484001 

657 

978777 

67 

5o5724 

724 

494276 

14 

47 

484895 
4852S9 

656 

978737 

6-: 

5o6i59 
506593 

724 

493841 

i3 

48 

655 

978696 

68 

723 

493407 

13 

49 

485682 

655 

978655 

68 

507027 

722 

492973 

II 

5o 

486075 

654 

978615 

68 

507460 

722 

492340 

10 

5i 

9-486467 

653 

9-978574 

68 

9-507893 

721 

10-492107 

I 

32 

486B60 

653 

978533 

68 

5o8326 

721 

491674 

53 

487251 

652 

978493 

68 

508759 

720 

49 1 241 

7 

54 

487643 

65 1 

978432 

68 

509191 

719 

490809 

6 

55 

488034 

65 1 

978411 

68 

509622 

719 

490378 

5 

56 

488424 

630 

978370 

68 

5ioo54 

718 

489946 

4 

57 

488814 

65o 

978329 

68 

5 I 0485 

718 

48901 5 

3 

58 

489204 

649 

978288 

68 

510916 
5ii346 

717 

4S90S4 

3 

59 

489593 

648 

978247 

68 

716 

4S8604 

1 

60 

489982 

648 

978206 

68 

511776 

716 

488224 

0 

'1 
107 

Cosine. 

D. 

Sine. 

D.  1 

Cotang. 

D. 

Tang. 

f 

0 

•J 

go 

36 

LOGARITHMIC  SINES 

,  TANGENTS,  ETC.    Table  IL  | 

^18^ 

> 

ICIO  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

f 

0 

9-489082 
490371 

648 

9-978206 

68 

9.611776 

716 

10-488224 

60 

I 

648 

978165 

68 

612206 

716 

487794 

60 

2 

490759 

647 

978124 

68 

5 1 2635 

716 

487366 

68 

3 

491 147 

646 

978083 

69 

6i3o64 

714 

486936 
486607 

57 

4 

491535 

646 

978042 

69 

613493 

714 

66 

5 

491922 

645 

978001 

69 

613921 

''■^ 

486079 

66 

6 

492308 

644 

977960 
977918 

69 

614349 

7>3 

486661 

54 

I 

492695 

U4 

59 

614777 

712 

485223 

53 

493081 

643 

977877 

69 

61 3204 

712 

484796 

62 

9 

493466 

642 

977835 

69 

5i563i 

711 

484369 
483943 

5i 

10 

493851 

642 

977794 

69 

616067 

710 

60 

II 

9-494236 

641 

9-977762 

69 

9.616484 

710 

10-483516 

49 

12 

494621 

641 

97771 1 

69 

516910 
517335 

709 

483090 

48 

i3 

495oo5 

640 

977660 
977628 

69 

700 

482666 

47 

14 

495388 

639 

69 

617761 

708 

482239 

46 

i5 

495773 

639 

977686 

69 

618186 

708 

481814 

45 

j6 

496154 

638 

977544 

70 

618610 

707 

481390 

44 

17 

496537 

637 

977603 

70 

619034 

706 

480966 

43 

i8 

496919 
497301 

637 

977461 

70 

619438 

706 

480342 

42 

•9 

636 

977419 

70 

619882 

706 

480118 

41 

20 

497682 

636 

977377 

70 

52o3o5 

705 

479695 

40 

21 

9-498064 

635 

9-977335 

70 

9.620728 

704 

10-479272 

39 

22 

498444 

634 

977293 

70 

621161 

7o3 

47«849 

38 

23 

498825 

634 

977231 

70 

621673 

703 

478427 

37 

24 

499204 

633 

977209 

70 

621996 

703 

478006 

36 

23 

499084 

632 

977J67 

70 

622417 

702 

477583 

36 

26 

499963 
5oo342 

632 

977125 

70 

622833 

702 

477162 

34 

27 

63i 

977083 

70 

623269 

701 

476741 

33 

28 

500721 

63 1 

977041 

70 

623680 

701 

476320 

32 

29 

601099 

63o 

976999 

70 

624100 

700 

476900 

3i 

3o 

501476 

629 

976937 

70 

624620 

699 

476480 

3o 

3i 

9-5oi854 

629 

9-976914 

70 

9-624940 

t^ 

10-476060 

29 

32 

50223l 

628 

976872 

71 

62535Q 

474641 

28 

33 

602607 

628 

976830 

7J 

626778 

'  698 

474222 

27 

34 

602984 

627 

976787 

7> 

626197 

697 

4738o3 

26 

35 

5o336o 

626 

976745 

71 

626616 

697 

473386 

26 

36 

603735 

626 

976702 

71 

627033 

696 

472967 

24 

37 

604110 

625 

976660 

71 

627461 

696 

472349 

23 

33 

604485 

625 

976617 

71 

627868 

696 

472132 

22 

39 

604860 

624 

976374 

11 

628285 

696 

471715 

21 

40 

5o5234 

623 

976532 

71 

628702 

694 

471298 

20 

41 

9-5o56o3 

623 

9-976489 

71 

9-629119 

693 

10-470881 

10 

42 

506981 

622 

976446 

71 

629633 

693 

470466 

18 

43 

5o6354 

622 

976404 

71 

529961 

693 

470049 

17 

44 

606727 

621 

976361 

71 

53o366 

692 

469634 

16 

45 

607099 

620 

976318 

71 

530781 

691 

469219 
468804 

i5 

46 

607471 

620 

976273 

7' 

531196 

691 

14 

47 

607843 

619 

976232 

72 

63i6ii 

690 

468389 

i3 

48 

,608214 

619 

976189 

72 

632023 

690 

467076 

12 

49 

6oS585 

618 

976146 

72 

532439 
532863 

689 

467361 

11 

5o 

608955 

6x8 

976103 

72 

689 

467147 

10 

5i 

9-609326 

6.7 

9-976060 

72 

9-533266 

683 

10-466734 

t 

52 

609696 

616 

976017 

72 

533679 

688 

466321 

53 

6ioo65 

616 

973974 

72 

534092 

687 

466908 

7 

54 

610434 

6i5 

976930 

72 

534604 

687 

466496 
466084 

6 

55 

5io8o3 

6i5 

976887 

72 

534916 

686 

5 

56 

611172 

614 

975844 

72 

635328 

686 

464672 

4 

57 

5ii54o 

6i3 

976800 

72 

635739 

685 

464261 

3 

58 

611907 

6i3 

976767 

72 

636 I 5o 

685 

463850 

a 

59 

612275 

612 

973714 

72 

536561 

684 

463439    I 

60 

612642 

612 

973670 

72 

636972 

684 

463023    0 

/ 

Cosine.   | 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang.     / 

108 

0 

71° 

Table  1 1. 

LOGARITHMIC 

SINES,  TANGENTS,  ETC. 

37] 

19° 

100°  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang, 

/ 
6c 

0 

g- 512642 

612 

9-975670 

73 

9-536972 
537382 

684 

io-463o28 

I 

5i3oo(^ 

611 

973627 

73 

683 

462618 

u 

2 

5i337J 

611 

975583 

73 

537792 

683 

462208 

3 

5i374i 

610 

975539 

73 

538202 

682 

461798 

57 

4 

5 1 4 1 07 

609 

975496 

73 

538611 

682 

461389 

56 

5 

5i447i 

609 
608 

975432 

73 

539010 

681 

460980 
460371 

55 

6 

5i4837 

975408 

73 

539429 

681 

54 

I 

5.1  3202 

608 

975365 

73 

539837 

680 

460163 

53 

5 1 5566 

607 

975321 

73 

540245 

680 

459755 

52 

9 

5i593o 

607 

975277 

73 

540653 

679 

439347 

5i 

10 

516294 

606 

975233 

73 

541061 

679 

458939 

5o 

!I 

9-5i6657 

6o5 

9-975189 
973145 

73 

9-541468 

678 

10-458532 

12 

13 

517020 

6o5 

73 

541875 

678 

458i25 

i3 

517382 

604 

975ioi 

73 

542281 

677 

457719 

47 

i4 

517745 

604 

975057 

73 

542688 

677 

437312 

46 

i5 

518107 
518468 

6o3 

975oi3 

73 

543094 

676 

436906 

45 

i6 

6o3 

974969 
974Q25 

74 

543499 

676 

456301 

44 

17 

518829 

602 

74 

543903 
544310 

675 

456093 

43 

i8 

5tqicp 

601 

974880 

74 

675 

455690 

42 

'9 

519331 

601 

974836 

74 

544715 

674 

455285 

41 

20 

519911 

600 

974792 

74 

545119 

674 

454881 

40 

21 

9-520271 

600 

9-974748 

74 

9-545524 

673 

10-454476 

ll 

22 

52o63i 

599 

974703 

74 

545928 
546331 

673 

454072 

23 

520990 

^2 

974659 

74 

672 

453669 

u 

24 

52  1 349 

598 

974614 

74 

546735 

672 

453263 

25 

521707 

598 

974570 

74 

547138 

671 

452862 

35 

26 

522066 

597 

974523 

74 

547540 

671 

452460 

34 

27 

522424 

596 

974481 

74 

547943 
548345 

670 

432037 

33 

28 

522781 

596 

974436 

74 

670 

431655 

32 

29 

523i38 

593 

974391 

74 

548747 

669 

451233 

3i 

3o 

523495 

593 

974347 

75 

549149 

669 

45o85i 

3o 

3i 

9-523852 

594 

9-974302 

75 

9-549550 

668 

io-45o45o 

?? 

32 

524208 

594 

974257 

75 

549951 
55o352 

668 

45oo4o 

33 

524564 

^9^ 

974212 

75 

667 

449648 

27 

3i 

524920 

593 

974167 

75 

550752 

667 

449248 

26 

35 

525275 

592 

974122 

75 

55ii53 

666 

448847 

25 

36 

525630 

591 

974077 

75 

55i552 

666 

448448 

24 

37 

525o84 
526339 

591 

974o32 

75 

55l()32 

665 

448048 

23 

38 

590 

9739S7 

75 

53235i 

665 

447649 

22 

39 

526693 

590 

973942 

75 

552750 

665 

447230 

21 

40 

527046 

589 

973897 

75 

553149 

664 

446851 

20 

41 

9-527400 

f^2 

9-973852- 

75 

9-553548 

664 

10-446452 

\t 

42 

527753 

588 

973807 

75 

553o46 
554344 

663 

446054 

43 

528io5 

588 

973761 

75 

663 

445656 

17 

44 

528458 

587 

973716 

76 

554741 

662 

443259 

16 

45 

528810 

587 

973671 

76 

555i39 

662 

444861 

l5 

46 

529161 

586 

973625 

76 

555536 

661 

444464 

14 

47 

5295i3 

586 

973580 

76 

555933 
556329 

661 

444067 

i3 

48 

529864 

585 

973535 

76 

660 

443671 

12 

49 

53o2i5 

585 

973489 

76 

556725 

660 

443275 

II 

5o 

53o565 

584 

973444  . 

76 

557121 

659 

442879 

10 

5i 

9-53o9i5 

584 

9-973398 

76 

9-557517 

639 

10-442483 

Q 

52 

531265 

583 

973332 

76 

557913 

639 

442087 

8 

53 

53i6i4 

582 

973307 

76 

6583o8 

658 

441692 

7 

54 

531063 
532312 

582 

973261 

76 

558703 

658 

441297 

6 

55 

58i 

973215 

76 

559097 

657 

440903 

5 

56 

532661 

58 1 

973169 

76 

559491 
559885 

657 
656 

440309 

4 

57 

533009 

58o 

973124 

76 

440113 

3 

58 

533357 

58o 

973078 

76 

560279 
560673 

656 

439121 

a 

59 

533704 

l]l 

973o32 

77 

655 

439327 

I 

6o 

534032 

972986 

77 

56 1066 

655 

438934 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

lOS 

° 

ro" 

38 

LOGARITHMIC  SINES, 

TANGENTS,  ETC.   Table 

TH 

20° 

159° 

/ 

0 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang.   j  ' 

9-534052 

578 

9.972986 

77 

9061066 

655 

10-438934 

60 

I 

534399 

577 

972940 

77 

561439 

654 

438341 

5o 

2 

534743 

577 

972S94 

7'' 

56i83i 

634 

438149 

58 

3 

535092 

577 

972848 

77 

562244 

653 

437756 

57 

4 

535433 

076 

972802 

77 

562636 

653 

437364 

56 

5 

535783 

576 

972755 

77 

563028 

653 

436972 

55 

6 

536129 

373 

972709 

77 

563419 

652 

436581 

54 

7 

536474 

^•^ 

972663 

77 

563811 

652 

436189 

53 

8 

536818 

574 

972617 

77 

564202 

65i 

435798 

52 

9 

537163 

573 

972370 

77 

564593 

65i 

435407 

5i 

10 

537507 

573 

972324 

77 

564983 

65o 

435017 

5o 

II 

9-537851 

572 

9.972478 

77 

9-565373 

65o 

10-434627 

49 

12 

538194 

572 

972431 

78 

563763 

649 

434237 

43 

i3 

538533 

571 

972383 

78 

566 1 53 

649 

433847 
433458 

47 

i4 

53^880 
539223 

57. 

972333 

78 

566542 

649 

46 

i5 

570 

972291 

78 

566032 
567320 

648 

433068 

45 

i6 

539365 

570 

972245 

78 

643 

432680 

44 

17 

539907 

569 

972198 

78 

567709 
56809S 

647 

432291 

43 

i8 

540249 

569 

972131 

78 

647 

431902 

42 

'9 

540390 

568 

972 1 o5 

78 

568486 

646 

43i3i4 

41 

20 

540931 

568 

972o58 

78 

568873 

646 

431127 

40 

21 

9-541272 

567 

9-972011 

78 

9-569261 

645 

10-430739 

39 

22 

54i6i3 

567 

971964 

78 

569643 

645 

43o352 

38 

23 

541953 

566 

9T9'7 

78 

570035 

645 

429965 

37 

24 

542293 

566 

971870 

78 

570422 

644 

429378 

36 

23 

542632 

565 

971823 

78 

570809 

644 

429191 

428805 

35 

26 

542971 

565 

971776 

78 

571 193 

643 

34 

11 

543310 

564 

971729 

79 

571581 

643 

428419 

33 

543649 

564 

971682 

79 

571967 

Ui 

428033 

32 

29 

5439S7 

563 

9Ti633 

79 

572332 

642 

427643 

•3 1 

3o 

544323 

5o3 

971588 

79 

572733 

642 

427262 

3o 

3i 

q- 544663 

562 

9-97i54o 

79 

9-573123 

641 

10-426877 

29 

32 

543000 

562 

971493 

79 

573307 

641 

426493 

23 

33 

545338 

56 1 

971446 

79 

573892 

640 

426108 

27 

34 

543674 

56 1 

971398 

79 

574276 

640 

425724 

26 

35 

54601 1 

56o 

97i33i 

79 

574660 

639 

425340 

25 

36 

546347 

56o 

97i3o3 

79 

573044 

639 

424956 

24 

37 

546683 

559 

971236 

79 

575427 

639 

424373 

23 

38 

547019 

339 

971208 

79 

575810 

638 

424190 

22 

39 

547354 

558 

971161 

79 

576193 

638 

423807 

21 

40 

547689 

533 

971113 

79 

576376 

637 

423424 

20 

41 

9 -548024 

557 

9-971066 

80 

9-576959 

637 

io-423o4i 

19 

42 

548359 

557 

971018 

80 

577341 

636 

422659 

18 

43 

548693 

556 

970970 

Bo 

577723 

636 

422277 

•7 

44 

549027 

556 

970922 

80 

578104 

636 

42 1 896 

16 

45 

549360 

555 

970874 

80 

578486 

635 

42i5i4 

i5 

46 

549693 

555 

970827 

80 

578867 
579248 

635 

421133 

14 

47 

550026 

534 

970779 

80 

634 

420752 

i3 

48 

55o359 

534 

970731 

80 

579629 

634 

420371 

12 

49 

55o6g2 

553 

970683 

80 

580009 

634 

419991 

II 

DO 

55io24 

553 

970635 

80 

580389 

633 

419611 

10 

5i 

9-551356 

552 

9-970586 

80 

9-580769 

633 

io-4io23i 

0 

32 

551687 

552 

970538 

80 

581149 

632 

4i885i 

8 

53 

552018 

532 

970490 

80 

58i528 

632 

418472 

7 

54 

552349 

531 

970442 

80 

581907 

632 

418093 

6 

55 

552680 

53i 

970394 

80 

582286 

63 1 

417714 

5 

56 

553oio 

55o 

970345 

81 

582665 

63 1 

417335 

4 

11 

553341 

55o 

970297 

81 

583044 

63o 

416956 

3 

553670 

549 

970249 

8i 

583422 

63o 

416378 

3 

59 

554000 

549 

970200 

81 

583800 

629 

416200 

I 

bi 

554329 

548 

970i52 

81 

584177 

629 

4i5823 

0 

1 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

1T( 

1° 

S9° 

Table  II.    LOGARITHMIC  SINES 

TANGENTS,  ETC.        39  | 

21° 

159* 

/ 

Sine. 

D. 

Cosine.   1 

D. 

Tang. 

D. 

Cotang.    / 

0 

Q.554329 

548 

9'970i52 

81 

9-584177 

629 

10.415823 

60 

I 

554658 

548 

970103 

81 

584553 

629 

628 

415445 

?? 

2 

554987 

547 

970055 

81 

584932 

4i5o68 

3 

5553i5 

547 

970006 

81 

585309 

628 

414691 

37 

4 

555643 

546 

969957 

81 

585686 

627 

4i43i4 

56 

5 

555971 

546 

969009 
969860 

81 

586062 

627 

413938 

55 

6 

556299 

545 

81 

58U39 

627 

4i336i 

54 

I 

556626 

545 

96981 1 

8i 

586813 

626 

4i3i85 

53 

556953 

5a 

969762 

&i 

587190 

626 

412810 

52 

9 

557280 

544 

969714 

81 

587566 

625 

412434 

31 

10 

557606 

543 

969665 

81 

587941 

625 

412059 

30 

II 

Q-557932 

543 

9-969616 

82 

9-588316 

625 

10-411684 

49 

48 

13 

558258 

543 

969567 

82 

588691 

624 

4ii3o9 

i3 

558583 

542 

969518 

82 

5890&6 

624 

410934 

47 

14 

558909 

542 

969469 

82 

589440 

623 

410360 

46 

13 

559234 

541 

969420 

82 

589814 

623 

410186 

43 

i6 

559558 

541 

969370 

82 

590188 

b2i 

409812 

44 

17 

559S83 

540 

969321 

82 

590562 

622 

409433 

43 

i8 

560207 

540 

969272 

82 

590935 
591308 

622 

409065 

42 

'9 

56o53 I 

539 

969223 

82 

622 

408692 

41 

20 

56o855 

539 

969173 

82 

591681 

621 

408319 

40 

21 

9-561173 

538 

g. 9691 24 

82 

9.592054 

621 

10-407946 

39 

33 

22 

56i5oi 

538 

969075 

82 

592426 

620 

407574 

23 

561824 

537 

969025 

82 

592799 

620 

407201 

37 

24 

562146 

537 

968976 

82 

593171 

619 

406829 

36 

20 

562468 

536 

968926 

83 

593542 

619 

406453 

35 

26 

562790 

536 

968877 

83 

593914 

618 

406086 

34 

U 

563II2 

536 

968827 

83 

594285 

618 

40571 5 

33 

563433 

535 

968777 
968728 

83 

594656 

618 

405344 

32 

29 

563755 

535 

83 

595027 

617 

404973 

3i 

3o 

564075 

534 

968678 

83 

595398 

617  . 

404602 

3o 

3i 

9.564396 

534 

g. 968628 

83 

q- 595768 

617 

10-404232 

29 

32 

564716 

533 

968578 

83 

596138 

616 

4o3862 

23 

33 

565o36 

533 

968528 

83 

596508 

616 

403492 

27 

34 

565356 

532 

968479 

83 

596878 

616 

4o3l22 

26 

35 

565676 

532 

968429 

83 

597247 

6i5 

402753 

25 

36 

565995 
5663 1 4 

53 1 

968379 

83 

597616 

6i5 

402384 

24 

37 

53 1 

968329 

83 

597985 

6i5 

4o2oi5 

23 

38 

566632 

53i 

968278 

83 

5o8354 

614 

401646 

22 

■39 

566951 

53o 

968228 

84 

598722 

614 

401278 

21 

40 

567269 

53o 

968173 

84 

599091 

6i3 

400909 

20 

41 

9.567587 

529 

9-968128 

84 

9.599459 

6i3 

io-4oo54i 

'9 

42 

567904 

539 

52§ 

968078 

84 

599827 

6i3 

400173 

18 

43 

563222 

968027 

84 

600194 

612 

399806 

17 

44 

568539 

523 

07977 

84 

6oo562 

612 

39943s 

16 

45 

568856 

528 

967027 

84 

600929 

611 

399071 

i5 

46 

569172 

527 

967876 

84 

601296 

611 

398704 

14 

^1 

569488 

527 

967826 

84 

601 663 

611 

398337 

i3 

4S 

569804 

526 

967775 

84 

602029 

610 

397971 

12 

49 

570120 

526 

967725 

84 

602393 

610 

397603 

11 

5o 

570435 

523 

967674 

84 

602761 

610 

397239 

10 

5i 

9.570751 

525 

9-967624 

84 

9-6o3i27 

609 

10-396873 

9 

52 

071066 

524 

967573 

84 

603493 

609 

396507 

8 

53 

57i38o 

524 

967522 

85 

6o3858 

609 

396142 

7 

54 

571695 

523 

967471 

85 

604223 

608 

393777 

6 

55 

572009 

523 

967421 

85 

604583 

608 

395412 

5 

56 

572323 

523 

967370 

85 

604953 

607 

393047 

4 

57 

572636 

522 

967319 

85 

6o53i7 

607 

394683 

3 

58 

572900 

522 

967268 

85 

6o5682 

607 

394318 

2 

59 

573263 

521 

967217 

85 

606046 

606 

393954 

I 

6o 

573573 

521 

967166 

85 

606410 

606 

393390 

0 

/ 

Cosine. 

1   ^• 

Sine. 

D. 

Cotang. 

P. 

Tang. 

111 

0 

68° 

13 


40 

LOGARITHMIC  SINES, 

TANGENTS,  ETC.   Table 

"J 

22° 

157°  1 

t 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9.573575 

521 

9.967166 

85 

9'6o64io 

606 

I0'393590 

60 

I 

573888 

520 

967115 

85 

606773 
607137 

606 

393227 

% 

2 

574200 

520 

967064 

85 

6o5 

392863 

3 

574512 

5.9 

967013 

85 

607500 

6o5 

392500 

57 

4 

574824 

5.9 

966961 

85 

607863 

604 

392137 

56 

5 

575i36 

5>9 

966010 

85 

608225 

604 

391775 

55 

6 

575447 

5i8 

966859 

85 

.6o8588 

604 

391412 

54 

7 

575758 

5i8 

966808 

85 

608950 

6o3 

39io5o 

53 

8 

576069 

5.7 

966756 

86 

609312 

6o3 

390688 

52 

9 

576379 

5i7 

966705 

86 

609674 

6o3 

3go326 

5i 

10 

576689 

5i6 

966653 

86 

6ioo36 

602 

389964 

5o 

II 

9.576999 
577309 

5i6 

9.966602 

86 

9.610397 

602 

10.389603 

49 

12 

5i6 

966550 

86 

610739 

f^zi 

389241 

48 

i3 

577618 

5i5 

966499 

86 

611120 

601 

388880 

47 

14 

577927 

5i5 

966447 

86 

61 1480 

601 

388520 

46 

i5 

578236 

5i4 

966395 

86 

61 1 841 

601 

388i59 

45 

i6 

578545 

5i4 

966344 

86 

D12201 

600 

387799 
387439 

44 

17 

578853 

5i3 

966292 

86 

6i256i 

600 

43 

i8 

579162 

5i3 

966240 

86 

612921 

600 

387079 

42 

'9 

579470 

5i3 

966188 

86 

613281 

599 

386719 

41 

20 

579777 

5l2 

966136 

86 

6i364i 

599 

386339 

40 

21 

9.580085 

5l2 

9.966085 

87 

9.614000 

598 

10-386000 

39 

22 

580392 

5ii 

966033 

87 

614359 

598 

385641 

38 

23 

580699 

5ii 

965981 

87 

614718 

598 

385282 

37 

24 

58ioo5 

5ii 

965928 

87 

6i5o77 

597 

384923 

36 

25 

58i3i2 

5io 

965876 

87 

615435 

597 

384565 

35 

26 

58i6i8 

5io 

965824 

87 

615793 

597 

384207 

34 

27 

581924 

509 

965772 

87 

6i6i3i 

596 

383849 

33 

28 

582229 

509 

965720 

87 

616309 

596 

383491 
383i33 

32 

29 

58253d 

509 

965668 

87 

616867 

596 

3i 

3o 

582840 

5o8 

9656i5 

87 

617224 

595 

382776 

3o 

3i 

9.533145 

5o8 

9-965563 

87 

9.617582 

593 

10-382418 

29 

32 

583449 

5o7 

965511 

87 

617939 

595 

382061 

28 

33 

583754 

5o7 

965458 

87 

618293 

594 

381705 

27 

34 

584058 

5o6 

965406 

87 

6i8632 

594 

38 1 348 

26 

35 

584361 

5o6 

965353 

88 

619008 

594 

380992 

25 

36' 

584665 

5o6 

965301 

!^ 

619364 

593 

38o636 

24 

37 

584968 

5o5 

965248 

88 

619720 

593 

380280 

23 

38 

585272 

5o5 

965195 

88 

620076 

593 

379924 

22 

39 

585574 

5o4 

965143 

88 

620432 

592 

379568 

21 

40 

585877 

5o4 

960090 

88 

620787 

592 

379213 

20 

41 

9  5861 79 

5o3 

9.965037 

88 

9.621142 

592 

10-378858 

10 

42 

586482 

5o3 

964984 

88 

621497 

591 

378503 

18 

43 

586783 

5o3 

96493 1 

88 

621832 

591 

378148 

17 

44 

587085 

502 

964879 

88 

622207 

590 

377793 
377439 

16 

45 

587386 

5o2 

964826 

88 

622561 

590 

i5 

46 

587688 

5oi 

964773 

88 

622915 

590 

377085 

14 

% 

587989 

5oi 

964720 

88 

623269 
623623 

589 

376731 

i3 

588289 

5oi 

964666 

89 

589 

376377 

12 

49 

588590 

5oo 

964613 

89 

623076 

589 

376024 

II 

5o 

588890 

5oo 

964560 

89 

624330 

588 

375670 

10 

5i 

9.589190 

499 

9.964507 

89 

9.624683 

588 

10-.375317 

Q 

52 

589489 

499 

964454 

89 

625o36 

588 

374964 

0 

53 

589789 

499 

964400 

89 

625388 

587 

374612 

7 

54 

590088 

498 

964347 

89 

625741 

587 

374259 

6 

55 

590387 

498 

964294 

89 

626093 

587 

373907 

5 

56 

5906S6 

497 

964240 

89 

626445 

586 

373355 

4 

57 

590984 

497 

964187 

89 

626797 

586 

373203 

3 

58 

591282 

497 

964133 

89 

627149 

586 

372831 

a 

59 

591580 

496 

964080 

89 

627501 

585 

372499 

I 

60 

591878 

496 

964026 

89 

627802 

585 

372143 

0 

/ 

Cojine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

112 

0 

( 

?7° 

Table  II.    LOGARITHMIC  SINES, 

TANGENTS,  ETC 

41 

23° 

156® 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang.    / 

0 

9.591878 

496 

9-964026 

89 

9  627863 

585 

10.372148 

60 

I 

5921-6 

495 

963972 

89 

628203 

685 

371797 
371446 

u 

3 

592473 

493 

9630J9 
963863 

89 

628554 

585 

3 

592770 

495 

90 

b28905 

584 

371095 

57 

4 

593067 

4y4 

963811 

90 

629255 

684 

370745 

56 

5 

593363 

494 

963767 

90 

629606 

683 

370394 

65 

6 

593669 

493 

963704 

90 

629966 
63o3o6 

683 

870044 

64 

I 

593955 

493 

963630 

90 

683 

369694 

53 

594231 

493 

963596 

90 

63o656 

683 

36o344 
368996 
368645 

52 

9 

594547 

492 

963543 

90 

63ioo5 

583 

5i 

10 

594842 

492 

963488 

90 

631355 

583 

60 

II 

9-593137 

491 

9-963434 

90 

9-631704 

582 

10-368296 

49 
48 

12 

593432 

491 

963379 

90 

632033 

58 1 

367947 

i3 

693727 

491 

963333 

90 

632402 

58 1 

367398 

47 

i4 

696021 

490 

963271 

90 

632730 

58 1 

367230 

46 

i3 

5g63i5 

490 

963217 

90 

633099 

58o 

366901 
366553 

45 

i6 

696609 
696903 

489 

963163 

90 

633447 

58o 

44 

17 

489 

963 1 oS 

9' 

633796 

680 

866206 

43 

i8 

697196 

489 

963064 

9' 

634143 

679 

365857 

42 

'9 

597490 

488 

962999 

91 

634490 
634838 

679 

3655io 

41 

20 

697783 

488 

962943 

9' 

579 

866162 

40 

21 

9-598075 

487 

9-962890 
962836 

9> 

9-635i85 

678 

io-8648i5 

39 

22 

598368 

487 

9' 

636533 

678 

864468 

88 

23 

598660 

487 

962781 

9' 

636879 

578 

364I2I 

37 

24 

598933 

486 

962727 

9' 

636226 

577 

868774 

86 

23 

599244 

486 

962672 

91 

636672 

577 

863428 

85 

26 

699336 

485 

962617 

9' 

636919 

577 

363o8i 

34 

27 

699827 

485 

962362 

9' 

637263 

577 

862735 

33 

28 

600118 

485 

962608 

91 

637611 

676 

362889 

32 

29 

600409 

484 

962453 

91 

637956 

676 

862044 

3i 

3o 

600700 

484 

962398 

92 

638302 

576 

861698 

80 

3i 

9  -  600990 

484 

9-962343 

92 

9-638647 

575 

10-861353 

29 

32 

601280 

483 

962288 

92 

638992 
639337 

675 

861008 

28 

33 

601670 

483 

962233 

92 

675 

860668 

27 

34 

601860 

482 

962178 

92 

639683 

574 

860818 

26 

35 

602160 

482 

962123 

92 

640037 

674 

359973 

25 

36 

602439 

483 

962067 

92 

6^0371 

574 

359629 

24 

ll 

602728 

481 

962012 

92 

640716 

573 

359284 

23 

6o3oi7 

481 

961967 

92 

641060 

573 

868940 

22 

39 

6o33o5 

481 

961902 

92 

641404 

673 

868396 
858253 

21 

40 

603594 

480 

961846 

92 

641747 

673 

20 

4i 

9-603882 

480 

9-961701 
961735 

92 

9-642091 
642434 

673 

10-357909 

19 

42 

604170 

479 

92 

673 

867366 

18 

43 

604457 

479 

961680 

92 

642777 

673 

867228 

17 

44 

604745 

479 
478 

961624 

93 

643120 

671 

856880 

16 

45 

6o5o32 

961669 
961613 

93 

643463 

671 

856537 

i5 

46 

6o53i9 

473 

93 

643806 

571 

356194 

14 

47 

606606 

478 

961438 

93 

644148 

670 

355853 

i3 

48 

606892 

477 

961403 

93 

644490 

570 

356510 

13 

49 

606179 

477 

961346 

93 

644832 

670 

866168 

11 

DO 

606463 

476 

961290 

93 

646174 

669 

864826 

10 

5i 

9-606761 

476 

9-961235 

93 

9-646616 

669 

10-864484 

I 

33 

607036 

476 

961170 
961123 

93 

645867 

669 

864143 

53 

607322 

476 

93 

646199 

669 

353801 

7 

54 

607607 

475 

961067 

93 

646640 

56§ 

353460 

6 

55 

607892 

474 

961011 

93 

646881 

563 

353119 

5 

56 

608177 

474 

960955 

93 

647222 

563 

352778 

4 

37 

608461 

474 

960899 
960843 

93 

647662 

667 

862438 

3 

58 

608743 

473 

9^ 

647903 

567 

862097 

a 

59 

609029 

473 

960786 

94 

648243 

667 
566 

861737 

I 

6o 

6093 1 3 

473 

960730 

94 

648683 

351417 

0 

/ 

/ 

Cosine. 

D. 

Sine. 

D 

Cotang. 

D, 

Tang. 

11 

5° 

66=  1 

42 

LOGARITHMIC  SINES 

TAXGENTS,  ETC.    Table  II.  | 

24° 

156°  1 

1 

Sine. 

D 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9-6o93i3 

4i3 

9-960730 

94 

9-648583 

566 

io-35i4i7 

60 

I 

609597 

4?2 

960674 

94 

648923 

566 

351077 

1? 

2 

609880 

472 

960618 

94 

649263 

566 

330737 

3 

610164 

472 

960561 

94 

649602 

566 

350398 

57 

4 

610447 

471 

96o5o5 

94 

649942 

565 

35oo58 

56 

5 

610729 

471 

960448 

94 

65o28i 

565 

349719 

55 

6 

611012 

470 

960392 

94 

630620 

565 

349380 

54 

7 

611294 

470 

960335 

94 

650959 

564 

349041 

53 

8 

611576 

470 

960279 

94 

631297 

564 

348703 

52 

9 

6ii858 

469 

960222 

94 

65 1 636 

564 

348364 

5i 

10 

6i2i4o 

469 

960165 

94 

65 I 974 

563 

348026 

5o 

II 

9-612421 

460 

9-960109 

95 

9-632312 

563 

10-347688 

49 

12 

612702 

468 

960032 

95 

652650 

563 

347350 

48 

i3 

612983 

468 

959995 

95 

652988 

563 

347012 

47 

14 

613264 

467 

959q38 

95 

653326 

562 

346674 

46 

ID 

613545 

467 

959882 

93 

653663 

562 

346337 

45 

i6 

6i3825 

467 

959825 

95 

654000 

562 

346000 

44 

17 

6i4io5 

466 

959768 

95 

654337 

56 1 

345663 

43 

i8 

614385 

466 

959711 

95 

654674 

56 1 

345326 

42 

»9 

614665 

466 

959654 

95 

655oii 

56 1 

344989 

41 

20 

614944 

465 

959596 

95 

655348 

56i 

344652 

40 

21 

9-6i5223 

465 

9-959539 

95 

9-655684 

56o 

10-344316 

39 

22 

6i55o2 

465 

959482 

95 

656o2o 

56o 

343980 

38 

s3 

6 1 57.8 1 

464 

959425 

95 

656356 

56o 

343644 

37 

24 

616060 

464 

959368 

93 

656692 

559 

343308 

36 

23 

6 16338 

464 

959310 

96 

657028 

559 

342972 

35 

26 

616616 

463 

959253 

96 

657364 

559 

342636 

34 

37 

616894 

463 

939195 

96 

657699 

559 

342301 

33 

28 

617172 

462 

939133 

96 

658o34 

553 

341966 

32 

29 

617430 

462 

939080 

96 

658369 

558 

34i63i 

3i 

3o 

617727 

462 

959023 

96 

•  658704 

558 

341296 

3o 

3 1 

9-618004 

461 

9-958963 

96 

'■'S^il 

558 

10-340961 

29 

32 

618281 

461 

958908 
958850 

96 

557 

34ot>27 

28 

33 

6i8558 

461 

96 

659708 

557 

340292 

27 

34 

618834 

460 

938792 

96 

660042 

557 

339953 

26 

35 

619110 

460 

958734 

66 

660376 

557 

339624 

25 

36 

619386 

460 

958677 

96 

660710 

556 

339290 

24 

37 
38 

619662 

459 

958619 

96 

661043 

556 

338937 

23 

619938 

459 

938361 

96 

661377 

556 

338623 

22 

39 

620213 

459 

9585o3 

97 

66I7IO 

555 

338290 
337957 

21 

40 

620488 

458 

958445 

97 

662043 

555 

2C 

4i 

9-620763 

458 

9-938387 

97 

9-662376 

555 

10-337624 

19 

42 

621038 

457 

938329 

97 

662709 

554 

337291 

18 

43 

62i3i3 

437 

958271 

97 

663042 

554 

336953 

17 

44 

621587 

457 

958213 

97 

663375 

554 

336625 

j6 

45 

621861 

456 

958134 

97 

663707 

554 

336293 

i5 

46 

622135 

456 

958096 

97 

664039 

553 

335961 

14 

47 

622409 

456 

958o38 

97 

664371 

553 

335629 

i3 

48 

622682 

455 

957979 

97 

664703 

553 

335297 

12 

49 

622956 

455 

937921 

■97 

665o35 

553 

334965 

n 

5o 

623229 

455 

957863 

97 

665366 

552 

334634 

10 

5i 

9-6235o2 

454 

9-957804 

97 

9-665698 

552 

10-334302 

I 

32 

623774 

454 

957746 

98 

666029 

552 

333971 

53 

624047 

454 

957687 

98 

666360 

55i 

333640 

7 

54 

624319 

453 

957628 

98 

666691 

55i 

333309 

6 

55 

624391 

453 

957370 

98 

667021 

55 1 

332979 

5 

56 

624863 

453 

957511 

98 

667352 

55i 

332648 

4 

37 

623135 

452 

937432 

93 

667682 

330 

3323i3 

3 

58 

625406 

432 

%t 

93 

6680 1 3 

530 

331987 

a 

59 

623677 

432 

98 

668343 

55o 

33i657 

I 

60 

625948 

45i 

937276 

98 

668673 

55o 

33i327 

0 

'     Cosine. 

D. 

Sine. 

P. 

Cotan;. 

D. 

Tang. 

1 

11- 

P 

es^' 

Table  II.   LOGARITHMIC  SINES, 

TANGENl 

S,  ETC. 

:n 

•25° 
/ 

154°  1 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9-625943 

45i 

9.957276 

98 

9.668673 

55o 

10-33l327 

60 

I 

626219 

45i 

937217 

98 

669002 

549 

330998 

u 

2 

626490 

45 1 

957138 

98 

669332 

549 

33o668 

3 

626760 

45o 

957099 

98 

669661 

549 

33o339 

57 

4 

627030 

45o 

937040 

98 

669991 

548 

330009 

56 

5 

627300 

430 

936981 

98 

670320 

548 

329680 

55 

6 

627570 

449 

93692 1 

99 

670649 

548 

329351 

54 

7 

627840 

449 

936862 

99 

670977 
671306 

548 

329023 

53 

8 

628109 

449 

936803 

99 

547 

328694 

52 

9 

628378 

448 

936744 

99 

671635 

547 

328365 

5i 

10 

628647 

448 

956684 

99 

671963 

547 

328037 

5o 

II 

9-628916 

457 

9-956625 

99 

9-672291 

547 

10-327709 

40 

48 

12 

629185 

447 

936566 

99 

672619 

546 

827381 

i3 

629453 

447 

936506 

99 

672947 

546 

327053 

47 

14 

629721 

446 

936447 

99 

673274 

546 

826726 

46 

i5 

629989 

446 

936387 

99 

678602 

546 

826893 

45 

i6 

630237 

446 

936327 

99 

673929 

545 

826071 

44 

17 

63o524 

446 

936268 

99 

674257 

545 

825743 

43 

i8 

630792 

445 

936208 

100 

674584 

545 

325416 

43 

19 

63ioD9 

445 

936148 

100 

674911 

544 

825089 
324763 

41 

20 

63i326 

445 

936089 

100 

675237 

544 

40 

21 

9-63i593 

444 

9-936029 

100 

9-675564 

544 

10-324436 

It 

22 

63 1859 

444 

933969 

100 

675890 

544 

824110 

23 

632I2D 

444 

955909 

ICO 

676217 

543 

828783 

37 

24 

632392 

443 

955849 

100 

676543 

543 

328457 

36 

25 

6326D8 

443 

955789 

100 

676869 

543 

823181 

35 

26 

632923 

U3 

955729 

100 

677194 

543 

822806 

34 

27 

633189 

442 

955669 

100 

677520 

542 

822480 

33 

28 

633454 

442 

935609 

100 

.677846 

542 

822154 

32 

29 

633719 

442 

955548 

100 

678171 

542 

821829 

3i 

3o 

633984 

44i 

955488 

100 

678496 

542 

82i5o4 

3o 

3i 

9-634249 

441 

9-955428 

lOI 

9-678821 

541 

I0-32II79 

It 

32 

6345i4 

440 

935368 

101 

679146 

541 

320854 

33 

634778 

440 

955307 

101 

679471 

541 

320529 

27 

34 

635042 

440 

935247 

101 

679795 

541 

320203 

26 

35 

6353o6 

439 

955186 

101 

680120 

540 

819880 

25 

36 

635570 

439 

955126 

lOI 

680444 

540 

819556 

24 

37 

635834 

439 

955o65 

lOI 

680768 

540 

319232 

23 

38 

63 6097 

438 

955oo5 

lOI 

681092 

540 

3 1 8go8 

22 

39 

636360 

438 

934944 

lOI 

681416 

539 

8 18584 

21 

40 

636623 

438 

954883 

lOI 

681740 

539 

318260 

20 

4i 

9-636886 

437 

9-954823 

lOI 

9-682063 

539 

10-317937 

19 

42 

637148 

437 

954762 

101 

682387 

539 

317618 

43 

637411 

437 

954701 

101 

682710 

538 

817290 

17 

44 

637673 

437 

954640 

101 

683o33 

538 

3 1 6967 

16 

45 

637935 

436 

954570 

101 

683356 

538 

816644 

i5 

46 

638197 

436 

954318 

102 

683679 

538 

816821 

14 

47 

638438 

436 

954457 

102 

684001 

537 

3 15999 

i3 

48 

638720 

435 

9343c6 

102 

684324 

537 

815676 

13 

49 

63S981 

435 

934335 

102 

684646 

537 

3i5354 

II 

5o 

639242 

435 

954274 

102 

684968 

537 

3i5o32 

10 

5i 

9 -639503 

434 

9-934213 

I«2 

9-685290 

536 

iO'3i47io 

? 

52 

639764 

434 

.  954152 

102 

685612 

536 

314388 

8 

53 

640024 

434 

934090 

102 

685934 

536 

3 1 4066 

7 

54 

640284 

433 

954029 

102 

686255 

536 

818745 

6 

55 

640544 

433 

953968 

102 

686577 
686898 

535 

818423 

5 

56 

640804 

433 

938906 

102 

535 

818102 

4 

u 

641064 

■432 

953845 

102 

687219 

535 

812781 

3 

641324 

432 

953783 

102 

687540 

535 

812460 

3 

59 

641583 

432 

953722 

io3 

687861 

534 

812139 
811818 

1 

6o 

641842 

43 1 

953660 

io3 

688182 

534 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

Tn 

0 

< 

;4=> 

44 

LOUAEITHMIC  SINES, 

TANGENTS,  ETC.    Tablk 

^ 

26" 

158°  1 

'  1    Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9-641842 

43 1 

9.953660 

io3 

9-688182 

534 

io.3ii8i8 

60 

I 

642101 

43 1 

953599 

io3 

688502 

534 

3II498 

59 

2 

642360 

43 1 

953537 

io3 

688823 

534 

311177 

58 

3 

642618 

43o 

953475 

io3 

689143 

533 

3 10857 

57 

4 

642877 

43o 

953413 

lo3 

689463 

533 

3io537 

56 

5 

643i35 

43o 

953352 

io3 

689783 

533 

310217 

55 

6 

643393 

43o 

953290 

lOJ 

690103 

533 

309897 

54 

I 

643600 

429 

953228 

lo3 

690423 

533 

309577 
309258 

53 

643908 

429 

953166 

.03 

690742 

532 

52 

9 

644165 

429 

953104 

io3 

691062 

532 

30S938 

5i 

10 

644423 

428 

953042 

io3 

691381 

532 

30-8619 

5o 

II 

9.644680 

428 

9-952980 

104 

9-691700 

53 1 

io.3o83oo 

49 

12 

644936 

428 

952918 

104 

692019 

53 1 

307981 

48 

i3 

645193 

427 

952855 

104 

692338 

53 1 

307662 

47 

14 

645430 

427 

952793 

104 

692656 

53 1 

307344 

46 

ID 

645706 

427 

952731 

104 

692975 

53 1 

307025 

45 

i6 

645962 

426 

952669 

104 

693293 

53o 

306707 

44 

n 

646218 

426 

952606 

104 

693612 

53o 

3o6388 

43 

i8 

646474 

426 

952544 

104 

693930 

53o 

306070 

42 

>9 

646729 

425 

952481 

104 

694248 

53o 

305732 

41 

20 

646984 

425 

952419 

104 

694566 

529 

303434 

40 

21 

9-647240 

425 

9952356 

104 

9.694883 

529 

io.3o5ii7 

39 

22 

647494 

424 

952294 

104 

695201 

529 

304799 

38 

23 

647749 

424 

952231 

104 

695518 

529 

304482 

37 

24 

648004 

424 

952168 

io5 

695836 

529 

3o4i64 

36 

25 

648258 

424 

952106 

io5 

696153 

528 

3o3347 

35 

26 

648512 

423 

952043 

io5 

696470 

528 

3o333o 

34 

11 

648766 

423 

951980 

io5 

696787 

528 

3o32i3 

33 

649020 

423 

951917 

io5 

697103 

528 

302897 

32 

29 

649274 

422 

951854 

io5 

697420 

527 

3o23So 

3i 

3o 

649527 

422 

951791 

io5 

697736 

527 

802264 

3o 

3i 

9.649781 

422 

9-951728 

io5 

9-698053 

527 

10.301947 

29 

32 

65oo34 

422 

95i665 

io5 

698360 

527 

3oi63i 

28 

33 

650287 

421 

951602 

io5 

698685 

526 

3oi3i5 

27 

34 

65o539 

421 

951539 

io5 

699001 

526 

3oog99 

26 

35 

650792 

421 

951476 

lo5 

699316 

526 

3oo684 

25 

36 

65io44 

420 

951412 

:o5 

699632 

526 

3oo368 

24 

37 

65i297 

420 

951349 

106 

699947 

526 

3ooo53 

23 

38 

65 1 549 

420 

951286 

106 

700263 

525 

299737 

22 

39 

65 1 800 

419 

951222 

106 

700578 

525 

299422 

21 

40 

652032 

419 

951159 

106 

700893 

525 

299107 

20 

4i 

9-652304 

419 

9-951096 
95io32 

106 

9.701208 

524 

10-298792 

10 

42 

652555 

418 

106 

70i523 

524 

298477 

18 

43 

6528o6 

418 

950968 

106 

701837 

524 

298163 

17 

44 

653o57 

418 

95ooo5 
950841 

106 

702152 

524 

297848 

16 

45 

6533o8 

418 

106 

702466 

524 

297534 

i5 

46 

653558 

417 

950778 

106 

702781 

523 

297219 

14 

47 

6538o8 

417 

950714 

106 

703095 

523 

296905 

i3 

48 

654059 

417 

95o65o 

106 

703409 

5s3 

296391 

12 

49 

654309 
654558 

416 

930586 

106 

703722 

523 

296278 

11 

5o 

416 

95o522 

107 

7o4o36 

522 

293964 

10 

5i 

9-654808 

416 

9-950458 

107 

9.704350 

522 

10-295650 

9 

52 

655o58 

416 

950394 
95o33o 

107 

704663 

522 

295337 

8 

53 

655307 

4i5 

107 

704976 

522 

295024 

7 

54 

655556 

4i5 

950266 

107 

703290 

522 

294710 

6 

55 

6558o5 

4i5 

950202 

107 

7o56o3 

521 

294307 

5 

56 

656o54 

414 

95oi38 

107 

705916 

521 

294084 

4 

57^ 

6563o2 

414 

950074 

107 

706228 

521 

293772 

3 

58 

656551 

414 

950010 

107 

706541 

521 

293459 

a 

59 

656799 

4i3 

949945 

107 

706854 

521 

293146 

I 

60 

657047 

4i3 

949881 

107 

707166 

520 

292834 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

lie 

0 

63° 

Table  II.   LOGARITHMIC  SINES, 

TANGENTS,  ETC. 

45 

27° 
/ 

0 

1. 

52° 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

9-657047 

4i3 

9-949881 

107 

0.707166 

Sao 

10.292834 

60 

I 

6572q5 

4i3 

949816 

107 

707478 

520 

292522 

59 

2 

637542 

412 

949752 

107 

707790 
70810J 

520 

292210 

58 

3 

657790 

412 

949688 

108 

520 

291898 

57 

4 

658o37 

412 

949623 

108 

708414 

5i9 

291586 

56 

5 

658284 

412 

949553 

108 

708726 

5.9 

291274 

55 

6 

658531 

411 

949494 

108 

709037 

019 

290963 

54 

7 

638778 

411 

949429 

108 

709349 

519 

290631 

53 

8 

659025 

411 

949364 

108 

709660 

519 

290340 

52 

9 

659271 

410 

949300 

108 

709971 

5i8 

2Q002O 

5i 

10 

659317 

410 

949235 

108 

710282 

5i8 

289718 

5o 

II 

9-659763 

410 

9-949170 

108 

9-710593 

5i8 

10. 2 8940 J 
289090 

8 

12 

660009 

409 

949 lOD 

108 

710904 

5i8 

i3 

660253 

409 

94QO40 

108 

711215 

5i8 

288785 

47 

14 

66o5oi 

409 

948975 

108 

711523 

5.7 

288475 

46 

i5 

660746 

409 

9489 1 0 

108 

711836 

5.7 

288164 

45 

i6 

660991 

4o3 

948845 

108 

712146 

5.7 

287854 

44 

n 

6612J6 

408 

948780 

109 

712436 

5.7 

287544 

43 

i8 

661481 

408 

948715 

109 

712766 

5i6 

287234 

42 

<9 

661726 

407 

948650 

109 

713076 

5i6 

286924 

41 

20 

661970 

407 

948584 

109 

713386 

5i6 

286614 

40 

21 

9-662214 

407 

9-948519 

109 

9-713696 

5i6 

10 -286304 

39 

38 

22 

662439 
662703 

407 

948454 

109 

714005 

5i6 

285995 

23 

406 

948388 

109 

714314 

5i5 

285686 

37 

24 

6629/(6 

406 

948323 

109 

714624 

5i5 

285376 

36 

23 

663190 
6634J3 

406 

948237 

109 

714933 

5i5 

283067 

35 

26 

4o5 

948192 

109 

715242 

5i5 

284753 

34 

27 

663677 

4o5 

948126 

109 

7i555i 

5i4 

284449 

33 

23 

663920 

4o5 

948060 

109 

71 5860 

5i4 

284140 

32 

29 

664163 

403 

947995 

110 

716168 

5i4 

283832 

3i 

3o 

664406 

404 

947929 

no 

716477 

5i4 

283523 

3o 

3i 

9-664648 

404 

9 -947863 

no 

9  716785 

5i4 

io.2832i5 

20 

28 

32 

664891 

404 

947797 
947731 

no 

717093 

5i3 

282907 

33 

665 1 33 

4o3 

no 

717401 

5i3 

282599 

27 

34 

665375 

4o3 

947665 

no 

717709 

5i3 

282291 

26 

35 

665617 

4o3 

947600 

no 

718017 

5i3 

281983 

25 

36 

665859 

402 

947333 

no 

718325 

5i3 

281675 

24 

37 

666100 

402 

947467 

no 

718633 

5l2 

281367 

23 

38 

666342 

402 

947401 

no 

718940 

5l2 

281060 

22 

39 

666583 

402 

947335 

no 

719248 

5l2 

280752 

21 

40 

666824 

401 

947269 

no 

719555 

5l2 

280445 

20 

41 

9-667065 

401 

9-947203 

no 

9-719862 

5l2 

io.28oi38 

:? 

42 

667305 

401 

9471 36 

III 

720169 

5n 

279831 

43 

667546 

401 

947070 

in 

720476 

5ii 

279524 

n 

44 

667786 

400 

947004 

in 

720783 

5ii 

279217 

16 

45 

668027 

400 

946937 

in 

721089 

5ii 

278911 

i5 

46 

668267 

400 

946871 

in 

721396 

5ii 

278604 

14 

47 

6685o6 

399 

946804 

in 

721702 

5io 

278298 

i3 

48 

668746 

399 

946738 

in 

722009 

5io 

277991 

12 

49 

668986 

399 

946671 

in 

7223l3 

5io 

277683 

II 

5o 

669225 

399 

946604 

in 

722621 

5io 

277379 

10 

5i 

9.669464 

398 

9-946538 

in 

9.722927 

5io 

10.277073 

t 

52 

669703 

3q8 

946471 

ni 

723232 

509 

276708 

53 

669942 

398 

946404 

in 

723533 

509 

276462 

7 

54 

670181 

397 

946337 

in 

723844 

509 

276156 

6 

55 

670419 

397 

946270 

112 

724149 

5o9 

273851 

5 

56 

670638 

397 

946203 

112 

724454 

509 
5o8 

275546 

4 

57 

670896 
671 i34 

397 

946i36 

112 

724760 

273240 

3 

58 

396 

946069 

112 

725o65 

5o8 

274933 

2 

59 

671372 

396 

946002 

112 

725370 

5o8 

274630 

1 

60 

671609 

396 

945935 

112 

725674 

5o8 

274326 
Tang. 

0 

/ 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

ir 

0 

^ 

52° 

46 

LOGARITHMIC  SINES, 

TAXGENTS,  ETC 

!.    Table  II, 

28° 

151° 

/ 

Sine. 

D. 

Cosine. 

D. 

Tung. 

D. 

Cutang. 

/ 

0 

9.671609 

396 

9-945935 

^  945868 

112 

9-725674 

5o8 

10-274326 

60 

I 

671847 

395 

112 

725979 

5o8 

274021 

?? 

2 

672084 

395 

945800 

112 

726284 

507 

273716 

3 

6-2321 

395 

945733 

112 

726588 

507 

273412 

57 

4 

672558 

395 

945666 

112 

726892 

5o7 

273108 

56 

5 

672795 
673032 

394 

945598 
945531 

112 

727197 

5o7 

272803 

55 

6 

394 

112 

727301 

507 

272499 

54 

7 

673268 

394 

945464 

Il3 

727805 

5o6 

272193 

53 

8 

673505 

394 

945396 

ii3 

728109 

5o6 

27189I 

52 

9 

673741 

393 

945328 

ii3 

728412 

5o6 

271588 

5i 

10 

673977 

393 

945261 

ii3 

728716 

5o6 

271284 

5o 

II 

9-674213 

393 

9-945193 

ii3 

9-729020 

5o6 

10' 270980 

^5 

12 

674448 

392 

945 1  25 

ii3 

729323 

5o5 

270677 

4S 

i3 

674684 

392 

945o58 

ii3 

729626 

5o5 

270374 

47 

14 

674919 

392 

944990 

ii3 

729929 

5o5 

270071 

46 

ID 

675i5d 

392 

9',4922 

ii3 

730233 

5o5 

269767 

43 

i6 

675390 

391 

944834 

ii3 

73o535 

5o5 

269465 

44 

17 

675624 

391 

9^47^6 

ii3 

73o838 

5o4 

269162 

43 

-1 8 

675859 

391 

944718 

ii3 

731141 

5o4 

268859 

42 

'9 

676094 

391 

944650 

ii3 

731444 

5o4 

268556 

41 

20 

676328 

390 

944582 

114 

731746 

5o4 

268254 

40 

21 

9-675562 

890 

9-944514 

1 14 

9-732048 

5o4 

10-267932 

39 
38 

22 

676796 

390 

944446 

114 

732351 

5o3 

267649 

23 

677030 

390 
389 

944377 

1 14 

732653 

5o3 

267347 

37 

24 

677264 

944309 

1 14 

732955 

5o3 

267045 

36 

25 

677498 
677731 

389 

944241 

1 14 

733257 

5o3 

266743 

35 

26 

389 

944172 

114 

733558 

5o3 

266442 

34 

27 

677964 

388 

944104 

1 1 4 

733860 

5o2 

266140 

33 

28 

678107 
678430 

388 

944o36 

1 14 

734162 

5o2 

265838 

32 

29 

388 

943967 

1 14 

734463 

502 

265537 

3i 

3o 

678663 

388 

943899 

1 14 

734764 

502 

265236 

3o 

3i 

9-678895 

387 

9-94383o 

114 

9  735066 

5o2 

10-264934 

20 

32 

679128 

387 

943761 

114 

735367 

502 

264633 

28 

33 

679360 

387 

943693 

ii5 

735668 

5oi 

264332 

27 

34 

679592 

387 

943624 

ii5 

735969 

5oi 

26403 1 

26 

35 

679824 

■  386 

943555 

ii5 

736269 

5oi 

263731 

25 

36 

68oo56 

386 

943486 

ii5 

736570 

5oi 

263430 

24 

37 

680288 

386 

943417 

ii5 

736870 

5oi 

263 !3o 

23 

38 

68o5i9 

385 

943348 

ii5 

737171 

5oo 

262829 

22 

39 

680750 

385 

943279 

ii5 

737471 

5oo 

262529 

21 

40 

680982 

385 

943210 

ii5 

737771 

5oo 

262229 

20 

4i 

9-68i2i3 

385 

9-943i4i 

ii5 

9-738071 

5oo 

10-261929 

10 

42 

681443 

384 

943072 

ii5 

738371 

5oo 

261629 

18 

43 

681674 

384 

943oo3 

ii5 

738671 

499 

261329 

17 

44 

681905 

384 

942934 

ii5 

738971 

499 

261029 

16 

45 

682135 

384 

942S64 

ii5 

739271 

499 

260729 

i5 

46 

682365 

383 

942795 

116 

739570 

499 

260430 

14 

47 

682595 

383 

942726 

116 

739870 

499 

260 i3o 

i3 

48 

682825 

383 

942656 

116 

740169 

499 

259831 

12 

49 

683o55 

383 

942587 

116 

740468 

498 

239332 

11 

5o 

683284 

382 

942517 

116 

740767 

498 

259233 

10 

5i 

9-6835i4 

382 

9-942448 

116 

9-741 066 

498 

10-258934 

I 

52 

683743 

382 

942378 

116 

741365 

498 

258635 

53 

683972 

382 

9423o8 

116 

741664 

498 

258336 

7 

54 

684201 

38i 

942239 

ii6 

741962 

497 

258o38 

6 

55 

684430 

38i 

942169 

116 

742261 

497 

257739 

5 

56 

684658 

38i 

942099 

116 

742559 

497 

23744 I 

4 

57 

684887 

38o 

942029 

116 

742858 

497 

257142 

3 

58 

685ii5 

38o 

941939 

116 

743 1 56 

497 

256844 

a 

59 

685343 

38o 

94 1 889 

"7 

743454 

497 

256546 

I 

60 

685571 

.  38o 

'941819 

"7 

743752 

496 

256248 

0 

/ 

Cosine, 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

]U 

i° 

61°  1 

Table  II.   LOGARITHMIC  SINES, 

TANGENTS,  ETC. 

7\ 

29° 
/ 

150°  1 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9-685571 

38o 

9.94l8l9 

117 

9-743752 

496 

10-256248 

60 

I 

685799 

379 

941749 

>'7 

744o5o 

496 

255930 

59 

2 

68O027 

379 

941679 

"7 

744348 

496 

255652 

58 

3 

686254 

379 

941609 

J'7 

744645 

496 

255355 

57 

4 

686482 

379 

378 

941539 

H7 

744943 

496 

255o57 

56 

5 

686709 

941469 

J>7 

745240 

496 

254760 

55 

6 

686936 

378 

941398 

>17 

745533 

495 

234462 

54 

I 

687163 

378 

941328 

U7 

745835 

495 

254165 

53 

687389 

378 

941258 

i'7 

746i32 

495 

253868 

52 

9 

6S7616 

.  377 

941187 

i'7 

746429 

495 

253571 

5i 

10 

687843 

377 

94M17 

117 

746726 

495 

253274 

5o 

II 

9-688069 

377 

9-941046 

118 

9-747023 

494 

10-252977 

% 

12 

68829:1 

377 

940975 

118 

74t3i9 

494 

252681 

i3 

688521 

376 

940905 
940834 

u8 

747616 

494 

252384 

47 

14 

68S747 

376 

118 

747913 

494 

252087 

46 

i5 

688972 

376 

940763 

118 

748209 

494 

251791 

45 

i6 

689198 

376 

940693 

118 

7485oD 

493 

251495 

44 

17 

689423 

375 

940622 

J18 

748801 

493 

251199 

43 

i8 

689648 

375 

94o55i 

118 

749097 

493 

25o9o3 

42 

19 

689873 

373 

940480 

118 

749393 

493 

250607 

41 

20 

690098 

375 

940409 

iiS 

749689 

493 

25o3ii 

40 

21 

9-690323 

374 

9-94o338 

iiS 

9-749985 

493 

io-25ooi5 

39 

22 

690548 

374 

940267 

118 

750281 

492 

249719 

38 

23 

690772 

374 

940196 

118 

730576 

492 

249424 

37 

24 

690996 

374 

940125 

119 

750872 

492 

249128 

36 

25 

691220 

373 

940054 

119 

751167 

492 

248833 

35 

26 

691444 

373 

939982 

119 

751462 

492 

248538 

34 

27 

691668 

373 

93991 1 

119 

751757 

492 

248243 

33 

28 

691892 

373 

939840 

"9 

752032 

491 

247948 

32 

29 

692115 

372 

939768 

119 

732347 

491 

247653 

3i 

3o 

692339 

372 

939697 

119 

752642 

491 

247353 

3o 

3i 

9-692562 

372 

9-939625 

119 

9-752937 

491 

10-247063 

g 

32 

692785 

37. 

939554 

119 

75323i 

491 

246769 

33 

693008 

371 

939482 

119 

753526 

491 

246474 

27 

34 

693231 

37  • 

939410 

119 

753820 

490, 

246180 

26 

35 

693453 

371 

939339 

119 

754n5 

490 

245885 

25 

36 

693676 

370 

939267 

120 

754400 
754703 

490 

245591 

24 

37 

693898 

370 

939193 

120 

490 

245297 

23 

38 

694120 

370 

939123 

120 

754997 

490 

245oo3 

22 

39 

694342 

370 

939052 

120 

755291 

490 

244709 

21 

40 

694564 

369 

938980 

120 

755585 

489 

244413 

20 

41 

9-694786 

369 

9-938908 

120 

9-755878 

489 

10-244122 

19 

42 

695007 

369 

938836 

120 

736172 

489 

243828 

18 

43 

695229 

369 

938763 

120 

756465 

489 

243535 

17 

44 

695430 

368 

938691 

120 

736759 

489 

243241 

16 

45 

695671 

368 

938619 

120 

757052 

489  , 

242948 

i5 

46 

695892 

368 

-938547 

120 

757345 

488 

242655 

14 

47 

696 1 1 3 

368 

938475 

120 

737633 

488 

242362 

i3 

43 

696334 

367 

938402 

121 

757931 

488 

242069 

12 

49 

696554 

367 

938330 

121 

758224 

483 

241776 

II 

DO 

696775 

367 

938258 

121 

738517 

483 

241483 

10 

5i 

9-696995 

357 

9-938i85 

121 

9-758810 

488 

10-241190 

9 

52 

697215 

366 

938ii3 

121 

75910a 

487 

240898 

8 

53 

697435 

366 

938040 

121 

759395 

487 

24o6o5 

7 

54 

697654 

366 

^^95 

121 

739687 

487 

24o3i3 

6 

53 

697874 

366 

121 

739979 

487 

240021 

5 

56 

698094 

365 

937822 

121 

760272 

487 

239728 

4 

u 

69S313 

365 

937749 

121 

760564 

487 

239',36 

3 

698532 

365 

937676 

121 

760856 

486 

239144 

2 

59 

698751 

365 

937604 

121 

761148 

486 

23^832 

I 

60 

698970 

364 

937531 

131 

761 4J9 

486 
D. 

238561 

0 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

Tang. 

1 

■3 

48 

LOGARITHMIC  SINES, 

TANGENTS,  ETC 

Table 

q 

30° 

149°  1 

/ 

Sine. 

D. 

Co4ne. 

D. 

,  Tang. 

D. 

Cotang. 

1 
60 

0 

9-698970 

364 

9'93753i 

121 

9.76i439 

486 

10.238561 

I 

699189 

364 

937458 

122 

761731 

486 

288269 

M 

2 

699407 

364 

937385 

122 

762023 

486 

287977 

3 

699626 

364 

987312 

122 

762314 

486 

287686 

57 

4 

699844 

363 

937238 

122 

762606 

485 

287894 

56 

5 

700062 

363 

937165 

122 

762897 

485 

287103 

55 

6 

700280 

363 

937092 

122 

763188 

485 

236812 

54 

I 

700498 

363 

937019 

122 

763479 

485 

236521 

53 

700716 

363 

936946 
936872 

122 

768770 

485 

286280 

52 

9 

700933 

362 

122 

764061 

485 

235989 

5i 

10 

7oii5i 

362 

936799 

122 

764352 

484 

235648 

5o 

11 

9-7oi368 

362 

9-936725 

122 

9-764643 

484 

10-235357 

49 

12 

70i585 

362 

936652 

123 

764933 

484 

235067 

48 

i3 

701802 

36i 

936578 

123 

765224 

484 

284776 
284486 

47 

i4 

702019 

36i 

9365o5 

123 

7655i4 

484 

46 

i5 

702  236 

36i 

936431 

123 

7658o5 

484 

284195 

45 

i6 

702452 

36i 

936357 

123 

766095 

484 

288905 

44 

n 

702669 

36o 

936284 

123 

766385 

483 

2336i5 

43 

i8 

702885 

36o 

936210 

123 

766675 

483 

233325 

42 

19 

7o3ioi 

36o 

9361 36 

123 

766965 

483 

233o35 

41 

20 

7o33i7 

36o 

936062 

123- 

767255 

483 

282745 

40 

21 

9-703533 

359 

9-935988 

123 

9-767545 

483 

10-232455 

39 

22 

703749 

359 

935914 

123 

767834 

483 

282166 

38 

23 

703964 

359 

935840 

123 

768124 

482 

281876 

S 

24 

704179 

359 

935766 

124 

768414 

482 

23 1 586 

25 

704395 

359 

935692 

124 

768703 

482 

281297 
23ioo8 

35 

26 

704610 

358 

935618 

124 

768992 

482 

34 

^^ 

704825 

358 

935543 

124 

769281 

482 

280719 

33 

7o5o4o 

358 

935469 
935393 

124 

769-371 

482 

280429 

32 

29 

705254 

358 

124 

769860 

481 

23oi4o 

3i 

3o 

705469 

357 

935320 

124 

770148 

481 

229852 

3o 

3i 

9-705683 

357 

9-935246 

124 

9-770437 

481 

10-229563 

29 

32 

705898 

357 

935171 

124 

770726 

481 

229274 

28 

33 

706112 

357 

935097 

124 

771013 

481 

228985 

27 

34 

706326 

356 

935022 

124 

77i3o3 

481 

228697 

26 

35 

706539 

'356 

934948 

124 

77092 

481 

228408 

25 

36 

706753 

356 

934873 

124 

771880 

480 

228120 

24 

37 

706967 

356 

934798 

125 

772168 

480 

227882 

23 

08 

707180 

355 

934723 

125 

772457 

480 

227543 

22 

39 

707393 

355 

934649 

125 

772745 

480 

227255 

21 

40 

707606 

355 

934574 

125 

773o33 

4S0 

226967 

20 

41 

9-707819 

355 

9-934499 

125 

9-773321 

480 

10-226679 

19 

42 

708032 

354 

934424 

125 

773608 

479 

226892 

18 

43 

708245 

354 

934349 

125 

773896 

479 

226104 

n 

44 

708458 

354 

934274 

125 

774184 

479 

2258i6 

16 

45 

708670 

354 

934199 

125 

774471 

479 

225529 

i5 

46 

708882 

353 

934123 

125 

774739 

479 

225241 

14 

s 

709094 

353 

934048 

125 

775046 

479 

224954 

i3 

709306 

353 

933973 

125 

775333 

479 

224667 

12 

49 

709518 

353 

933S98 

126 

775621 

478 

224879 

11 

5o 

709730 

353 

933822 

126 

775908 

478 

224092 

10 

5i 

9-709941 

352 

9  933747 

126 

9-776195 

478 

io-2238o5 

I 

52 

710153 

352 

933671 

126 

776482 

478 

228318 

53 

710364 

352 

933596 

126 

776768 

478 

228282 

7 

54 

710575 

352 

933520 

126 

777055 

478 

222945 

6 

55 

710786 

35i 

933445 

126 

777342 

478 

222658 

5 

56 

710997 

35i 

933369 
933293 

126 

777628 

477 

222872 

4 

u 

711208 

35i 

126 

777915 

477 

222085 

3 

711419 

35i 

933217 

126 

778201 

477 

221799 

3 

59 

711629 

35o 

933141 

126 

778488 

477 

22l5l2 

I 

60 

711839 

35o 

933066 

126 

778774 

477 

221226 

0 

'    Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

]'2( 

)° 

59° 

Table  II.    LOGARITHMIC  SINES, 

TANGENTS,  ETC 

49 

«1» 

148° 

f 

Sine. 

D. 

Cosine.   1 

D. 

Tang. 

D. 

Cotang. 

f 

0 

9»7Il839 

35o 

9-933066 

126 

9-778774 

477 

IO-22I326 

60 

I 

7i2o5o 

35o 

932990 

127 

779060 

477 

220940 

u 

2 

712260 

35o 

932914 
932838 

127 

779346 

476 

220654 

3 

712469 

349 

127 

77963a 

476 

J2o368 

57 

4 

712679 

349 

932762 

127 

779918 

476 

220082 

56 

5 

712880 

349 

932685 

127 

780203 

476 

219797 

55 

6 

713098 

349 

932609 
932533 

127 

780489 
780775 

476 

219511 

54 

I 

7i33o8 

349 

127 

476 

219225 

53 

713517 

348 

932437 

127 

781060 

476 

218940- 

52 

9 

713716 

348 

932380 

127 

781346 

475 

218654 

5i 

10 

713935 

348 

932304 

127 

78i63i 

475 

218369 

5o 

II 

9-714144 

34S 

9-932228 

127 

9-781916 

475 

io-2i8o84 

S 

13 

714352 

347 

932i5i 

\ll 

782201 

475 

217799 

i3 

714361 

347 

932075 

782486 

475 

217514 

47 

14 

714760 

347 

931998 

128 

782771 

475 

217229 

46 

i5 

714978 

347 

93IQ2I 

128 

783o56 

475 

216944 

45 

i6 

7i5i86 

^^7 

931845 

128 

783341 

475 

216609 

44 

•7 

715394 

346 

931768 

128 

783626 

474 

216874 

43 

i8 

7i56o2 

346 

931691 

128 

783910 

474 

216090 

42 

'9 

713809 

346 

931614 

128 

784195 

474 

2i58o5 

41 

20 

716017 

346 

931537 

128 

784479 

474 

2l552I 

40 

21 

9-716224 

345 

9-931460 

128 

9-784764 

474 

io-2i5236 

39 

22 

716432 

345 

93i383 

123 

785048 

474 

214952 

38 

23 

716639 

345 

93i3o6 

128 

785332 

473 

214668 

37 

24 

716846 

345 

931229 

129 

785616 

473 

214384 

36 

25 

717053 

345 

93ii52 

129 

785900 

473 

214100 

35 

26 

717259 

344 

931075 

129 

786184 

473 

2i38i6 

34 

11 

717466 

344 

930998 

129 

786468 

473 

213532 

33 

717673 

344 

930921 

129 

786752 

473 

213243 

32 

29 

717879 

344 

930843 

129 

787036 

473 

212964 

3i 

3o 

71808D 

343 

930766 

129 

787319 

472 

212681 

3o 

3i 

9-718291 

343 

9-930688 

129 

9-787603 

472 

10-212397 

g 

32 

718497 

343 

93061 1 

129 

787886 
788170 

472 

212114 

33 

718703 

343 

93o533 

129 

472 

2ii83o 

27 

34 

718909 

343 

930456 

129 

■788453 

,  472 

2 1 1 547 

26 

35 

719114 

342 

930378 

129 

788736 

472 

211264 

25 

36 

719320 

342 

93o3oo 

i3o 

789019 

472 

210981 

24 

37 

719525 

342 

930223 

i3o 

789302 

471 

210698 

23 

38 

719730 

342 

930145 

i3o 

789535 

471 

2io4i5 

22 

39 

719935 

341 

930067 

i3o 

789868 

471 

2IOl32 

21 

40 

720140 

341 

929989 

i3o 

790i5i 

471 

209849 

20 

41 

9-720345 

341 

9-929911 

i3o 

9-790434 

471 

10-209566 

]p 

42 

720549 

341 

929S33 

i3o 

790716 

471 

209284 

18 

43 

720754 

340 

929755 

i3o 

790999 

471 

20Q001 
208719 

17 

44 

720953 

340 

929677 

i3o 

791281 

471 

16 

45 

721162 

340 

929399 

i3o 

791563 

470 

208437 

i5 

46 

721366 

340 

929521 

i3o 

791846 

470 

2081 54 

14 

47 

721070 

340 

929442 

i3o 

792128 

470 

20787a 

i3 

48 

721774 

339 

929364 

i3i 

792410 

470 

207590 

12 

49 

721973 

339 

929286 

i3i 

792692 

470 

207368 

II 

5o 

722181 

339 

929207 

i3i 

792974 

470 

207026 

10 

5i 

c-722385 

339 

9-929129 

i3i 

9-793256 

470 

10-206744 

Q 

52 

722583 

339 
338 

929050 

i3i 

793538 

469 

206462 

8 

53 

722791 

928972 

i3i 

793819 

469 

206181 

7 

54 

722994 

338 

928893 

l3i 

794101 

469 

205899 

6 

55 

723197 

338 

928815 

i3i 

794383 

469 

2o56i7 

5 

56 

723400 

338 

92S736 

i3i 

794664 

469 

205336 

4 

57 

7236o3 

337 

92S657 

i3i 

794946 

469 

2o5o34 

3 

53 

]   7238o5 

337 

928378 

i3i 

795508 

469 

204773 

2 

59 

1   724007 

337 

928499 

i3i 

468 

204492 

1 

60 

724210 

337 

928420 

i3i 

795789 

468 

304211 

0 

/ 

' 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

12 

1° 

LS"   1 

50 

LOGARITHMIC  SINES, 

rANGENTS,  ETC 

Table  II. 

32° 

147=> 

t 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

60 

0 

9-724210 

i3r 

9.928420 

l32 

9.795789 

468 

I0«2O42ll 

I 

724412 

337 

928342 

l32 

79607c 

468 

208980 

% 

2 

724614 

336 

028263 

l32 

796351 

468 

208649 

3 

724816 

336 

928183 

l32 

796632 

468 

208368 

57 

4 

725017 

336 

928104 

l32 

796913 

468 

208087 

56 

5 

725219 

336 

928025 

l32 

797194 

468 

202806 

55 

6 

725420 

335 

927046 

i3a 

797474 

468 

202326 

54 

I 

725622 

335 

927867 

l32 

797755 

468 

202245 

53 

725823 

335 

927787 

l32 

798086 

467 

201964 

52 

9 

726024 

335 

927708 

l32 

798816 

467 

201684 

5i 

10 

726225 

335 

927629 

l32 

798596 

467 

201404 

5o 

11 

9-726426 

334 

9-927549 

l32 

9-798877 

467 

10-201128 

49 

12 

726626 

334 

927470 

1 33 

799157 

467 

200843 

48 

i3 

726827 

334 

927390 

i33 

799437 

467 

2oo563 

47 

14 

727027 

334 

927310 

i33 

799717 

467  . 

200288 

46 

i5 

727228 

334 

927231 

i33 

799997 

466 

2oooo3 

45 

i6 

727428 

333 

927151 

1 33 

800277 

466 

199723 

44 

17 

727628 

333 

927071 

i33 

800557 

466 

199443 

43 

i8 

727828 

333 

926991 

i33 

800886 

466 

199164 

42 

'9 

728027 

333 

926^11 

i33 

80U16 

466 

198884 

41 

20 

728227 

333 

926331 

i33 

801896 

456 

198604 

40 

21 

9-728427 
728626 

332 

9-936751 

i33 

9-801675 

466 

10-198825 

39 

22 

332 

926671 

i33 

801955 

466 

198045 

38 

23 

728825 

332 

926391 

i33 

802284 

465 

197766 

37 

24 

729024 

332 

926511 

i34 

8025 1 3 

465 

197487 

36 

.  25 

729223 

33i 

926431 

i34 

802792 

465 

197208 

35 

26 

729422 

33 1 

926351 

i34 

808072 

465 

196928 

34 

27 

729621 

33i 

926270 

i34 

8o335i 

465 

1 96649 

33 

28 

729820 

33i 

926190 

i34 

8o363o 

465 

196870 

32 

29 

780018 

33o 

926110 

134 

808909 

465 

1 9609 1 

3r 

3o 

730217 

33o 

926029 

1 34 

804187 

465 

195818 

3o 

3i 

9.730415 

33o 

9-925049 

923868 

i34 

9-804466 

464 

10-195534 

20 

32 

73o6i3 

33o 

i34 

804745 

464 

195255 

28 

33 

730811 

33o 

925788 

i34 

8o5o23 

464 

194977 

27 

34 

731009 

329 

923707 

i34 

8o58o2 

464 

194698 

26 

35 

731206 

329 

923626 

i34 

8o558o 

464 

194420 

25 

36 

73i4o4 

329 

925545 

i35 

8o5839 

464 

194141 

24 

37 

731602 

329 

925465 

i35 

806187 

464 

198868 

23 

3S 

731799 

329 

925384 

i35 

80641 5 

463 

198585 

22 

39 

731996 

328 

9253o3 

i35 

806693 

468 

198807 

2! 

40 

732193 

328 

923222 

i35 

806971 

468 

198029 

20 

41 

9-7323QO 

328 

9-925141 

i35 

9-807249 

463 

10-192751 

\% 

42 

732587 

328 

925060 

i35 

807527 

463 

192473 

43 

732784 

328 

924079 

i35 

807805 

463 

192195 

17 

44 

732980 

327 

924897 

i35 

808088 

463 

191917 

16 

45 

733177 

327 

924816 

i35 

8o836i 

463 

191689 

13 

46 

733373 

327 

924735 

i36 

808688 

462 

191362 

i4 

8 

733569 

327 

924654 

i36 

808916 

462 

191084 

i3 

73376D 

327 

924372 

i36 

809198 

462 

190807 

12 

49 

733961 

326 

924491 

i36 

80947 1 

462 

190529 

11 

5o 

734137 

326 

924409 

i36 

809748 

462 

190232 

10 

5i 

9-734353 

326 

9-924328 

1 36 

9-810025 

462 

10-189975 

0 

52 

734549 

326 

924246 

i36 

810802 

462 

189698 

8 

53 

734744 

325 

924164 

i36 

8io58o 

462 

189420 

7 

54 

734939 

323 

924083 

1 36 

810857 

462 

189143 

6 

55 

735i35 

325 

924001 

1 36 

8iii34 

461 

188866 

5 

56 

735330 

325 

923919 

1 36 

811410 

461 

188590 

4 

57 

735525 

325 

923837 

i36 

811687 

461 

18881 3 

3 

58 

735719 

324 

923755 

137 

811964 

461 

i88o36 

2 

59 

735914 

324 

923673 

:37 

812241 

461 

187759 

1 

6o 

736109 

324 

923591 

'37 

812317 

461 

187483 

0 

/ 

~ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

N 

2° 

57°  ! 

Table  II.    LOGARITHMIC  SINES, 

TANGENTS,  ETC 

^ 

83* 

• 

146°  1 

c 

Sine. 

D. 

Cosine. 

D. 

Tang. 

«• 

Coting. 

1 

o-736i09 
7363o3 

324 

9-923591 

1 37 

9-812517 

461 

10-187483 

60 

I 

324 

923509 

i37 

812794 

461 

187206 

% 

2 

736498 

324 

923427 

137 

8i3o70 

461 

186930 

3 

736692 

323 

923345 

i37 

813347 
8 1 3623 

460 

186653 

57 

4 

736886 

323 

923263 

i37 

460 

186377 

56 

5 

737080 

323 

923181 

137 

813899 

460 

186101 

55 

6 

737274 

323 

923098 

137 

814176 

460 

185824 

54 

T 

737467 

323 

923oi6 

i37 

814452 

460 

185548 

53 

^ 

737661 

322 

922933 
922851 

137 

814728 

460 

185272 

52 

9 

737855 

322 

i37 

8i5oo4 

460 

184996 

5i 

10 

738048 

322 

922768 

i38 

815280 

460 

184720 

5o 

II 

9-738241 

322 

9-922686 

i38 

9-8i5555 

459 

10-184443 

49 

12 

738434 

322 

•  922603 

i38 

8i583i 

459 

184169 
183893 

48 

i3 

738627 

321 

922520 

1 38 

816107 

459 

47 

i4 

738820 

321 

922438 

1 38 

8i6382 

459 

i836i8 

46 

i5 

739013 

321 

922355 

1 38 

8i6658 

459 

183342 

45 

:6 

739206 

321 

922272 

i38 

816933 

459 

183067 

44 

17 

739398 

321 

922189 

i38 

817209 

459 

182791 

43 

i8 

739590 

320 

922106 

i38 

817484 

459 

182516 

42 

'9 

739783 

320 

922023 

1 38 

817759 

459 

182241 

41 

20 

739975 

320 

921940 

i38 

8i8o33 

458 

181965 

40 

21 

9-740167 

320 

9-921857 

.39 

c-8i83io 

458 

10-181690 

S 

22 

740359 

320 

921774 

139 

8i8585 

458 

i8i4i5 

23 

74o55o 

3i9 

921691 

139 

818860 

458 

181 140 

37 

24 

740742 

3.9 

921607 

139 

819135 

458 

i8o865 

36 

25 

740934 

319 

921524 

139 

819410 

458 

180590 

35 

26 

741125 

319 

921441 

i39 

819684 

458 

i8o3i6 

34 

11 

74i3i6 

319 

3i8 

921357 

139 

819959 

458 

180041 

33 

74i5oS 

921274 

i39 

820234 

458 

179766 

32 

29 

741699 

3i8 

92 1190 

139 

82o5o8 

457 

17^492 

3i 

3o 

741889 

3i8 

921107 

139 

820783 

457 

179217 

3o 

3i 

9-742oSo 

3i8 

9-921023 

i3g 

9-821057 

457 

10-178943 

29 

32 

742271 

3i8 

920939 

140 

821332 

457 

178668 

28 

33 

742462 

317 

920856 

140 

821606 

457 

178394 

27 

34 

742602 

317 

920772 

1 40 

82188c 

457 

178120 

26 

35 

742842 

317 

920688 

140 

822x54 

457 

177846 

25 

36 

743o33 

3.7 

920604 

140 

822429 
82270J 

457 

177371 

24 

3- 

743223 

3i7 

920520 

140 

457 

177297 

23 

38 

743413 

3i6 

920436 

140 

822977 

456 

177023 

22 

3g 

743602 

3i6 

920352 

140 

823251 

456 

176749 

21 

40 

743792 

3i6 

920268 

140 

823524 

456 

1764-6 

20 

4i 

9-743982 

3i6 

9-920184 

140 

9-823798 

456 

10-176202 

10 

42 

7^4171 

3i6 

920099 

140 

824072 

456 

173928 

18 

43 

744361 

3i5 

920013 

140 

824345 

456 

175655 

17 

44 

744550 

3i5 

919931 

141 

824619 
824893 

456 

175381 

16 

45 

744739 

3i5 

919846 

141 

456 

173107 

15 

46 

744928 

3i5 

919762 

141 

825166 

456 

174834 

14 

47 

745  in 

3i5 

919677 

141 

825439 
825713 

455 

174561 

i3 

48 

745306 

3i4 

919593 

141 

455 

174287 

12 

49 

745494 

3i4 

919508 

141 

•  825986 

455 

174014 

u 

DO 

745683 

3i4 

919424 

141 

826259 

455 

173741 

10 

5i 

9- 74587 1 

3i4 

9-919339 

141 

9-826532 

455 

10-173468 

9 

32 

746060 

3i4 

919254 

141 

826803 

455 

173195 

8 

53 

746248 

3i3 

91916^ 

MI 

827078 

455 

172922 

7 

54 

746436 

3i3 

919083 

141 

827351 

455 

172649 

6 

55 

746624 

3i3 

919000 

141 

827624 

455 

1723-6 

5 

56 

746812 

3i3 

918915 

142 

827897 

454 

172103 

4 

57 

746999 

3i3 

9i883o 

142 

828170 

454 

171830 

3 

58 

747 '87 

3l2 

918745 

142 

828442 

454 

17 1 558 

a 

59 

747374 

3l2 

•918659 

142 

828715 

434 

171285 

I 

60 
/ 

747362 

3l3 

918574 

142 

828987 

454 

171013 

0 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

125 

0 

66° 

52 

LOGARITHMIC  SINES, 

TANGENTS,  ETC 

!.   Table 

^ 

34° 

145"  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9-747562 

3l2 

Q. 918574 

142 

9-828987 

454 

IO-171013 

60 

1 

747749 

3X2 

918489 

142 

829260 

454 

170740 

1% 

2 

747936 

3l2 

918404 

142 

829532 

434 

170468 

58 

3 

748123 

3ii 

9l83l8 

142 

829805 

434 

170195 

57 

4 

7483 10 

3ii 

918233 

142 

830077 

454 

169923 

56 

5 

748497 

3ii 

918147 

142 

83o349 

453 

169651 

55 

6 

748683 

3ii 

918062 

142 

83o62i 

453 

169379 

54 

7 

748870 

3ii 

917076 
91789I 

143 

830893 

453 

169107 

53 

8 

749056 

3io 

143 

83ii65 

453 

168835 

52 

9 

749243 

3io 

917805 

143 

83i437 

453 

168563 

5i 

10 

749429 

3io 

9'77i9 

143 

831709 

453 

168291 

5o 

II 

9-749615 

3io 

9-917634 

143 

9-831981 

453 

10-168019 

^ 

12 

749801 

3io 

917548 

143 

832253 

453 

167747 

i3 

749987 

3og 

917462 

143 

832523 

453 

167475 

47 

14 

730172 

309 

917376 

143 

332796 

453 

167204 

46 

i5 

75o358 

309 

917290 

143 

333o68 

452 

166932 

45 

i6 

75o543 

309 

917204 

143 

833339 

452 

166661 

44 

17 

730729 

309 

917118 

144 

833611 

452 

166389 
166118 

43 

i8 

750914 

3o§ 

917032 

144 

833882 

432 

42 

•9 

751099 

3o8 

916946 

144 

8341 54 

452 

165846 

41 

20 

751284 

3o3 

916859 

144 

834425 

452 

165573 

40 

21 

9-751469 

3o8 

9-916773 

144 

9-334696 

452 

io-i653o4 

ll 

22 

75i654 

3o8 

916687 

144 

334967 

452 

1 63033 

23 

751839 

3o8 

916600 

144 

835238 

452 

164762 

37 

24 

75202.3 

307 

9i65i4 

144 

335509 

432 

164491 

36 

23 

752208 

3o7 

916427 

144 

835780 

45i 

164220 

35 

26 

752392 

3o7 

916341 

144 

836o5i 

45i 

163949 

34 

27 

752576 

3o7 

916254 

144 

836322 

45i 

163678 

33 

28 

752760 

307 

916167 

145 

336593 

45i 

163407 

32 

29 

752944 

3o6 

916081 

145 

836864 

45 1 

i63i36 

3i 

3o 

753128 

3o6 

913994 

145 

837134 

45 1 

162866 

3o 

3i 

9-753312 

3o6 

9-915907 
915820 

145 

9-8374o5 

45i 

10-162595 

29 

32 

753495 

3o6 

145 

837675 

45i 

162325 

28 

33 

753679 

3o6 

913733 

143 

837946 

45i 

162054 

27 

34 

753862 

3o5 

915646 

145 

838216 

45i 

161784 

26 

35 

754046 

3o5 

915559 

145 

838487 

430 

i6i5i3 

23 

36 

754229 

3o3 

915472 

145 

838757 

45o 

161243 

24 

37 

754412 

303 

9i5385 

145 

339027 

45o 

160973 

23 

38 

754393 

3o5 

915297 

145 

839297 

45o 

160703 

22 

39 

754778 

3o4 

9i52io 

145 

839568 

430 

160432 

21 

40 

754960 

3  04 

9i5i23 

146 

839838 

45o 

160162 

20 

41 

9-755143 

3o4 

9 -915035 

146 

9-840108 

45o 

10-159892 

19 
18 

42 

755326 

3o4 

914948 

146 

340378 

45o 

159622 

43 

755508 

3  04 

914060 

146 

340648 

45o 

,59352 

17 

44 

755690 

3o4 

914773 
914685 

146 

340917 

449 

159083 

16 

45 

755872 

3o3 

146 

341187 

449 

i588i3 

i5 

46 

756o54 

3o3 

914398 

146 

841457 

449 

158543 

14 

8 

756236 

3o3 

914310 

146 

841727 

449 

158273 

i3 

756418 

3o3 

914422 

146 

841996 

449 

1 58004 

12 

49 

756600 

3o3 

914334 

146 

842266 

449 

157734 

11 

5o 

756782 

302 

914246 

147 

842535 

449 

157465 

10 

5i 

9-756963 

.302 

9-914158 

147 

9-842805 

449 

10-157195 

t 

52 

757144 

302 

914070 

147 

343074 

449 

156926 

53 

757326 

302 

913982 

147 

843343 

449 

156657 

1 

54 

757507 
757688 

302 

913894 

147 

843612 

449 
448 

156388 

6 

55 

3oi 

9i38o6 

147 

343882 

i56ii8 

5 

56 

757869 

3oi 

913718  . 

147 

8441 5i 

448 

155849 

4 

ll 

758o5o 

3oi 

9i363o 

147 

844420 

448 

155580 

3 

758230 

3oi 

913541 

147 

844689 

448 

i553ii 

3 

59 

75841 1 

3oi 

913453 

J  47 

844958 

448 

i55o42 

1 

60 

753591 

3oi 

9i3365 

147 

345227 

448 

154773 

0 

'    Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

12^ 

to 

550 

Tab 

LE  II.    LOGARITHMIC  SLNES, 

TAXGENTS,  ETC 

63  1 

86« 

114"  1 

'  1 

Sine. 

D.  ! 

Cosine. 

!>•  1 

Tang.    1 

D. 

Cotang. 

1 

0 

9-758591 

3oi  1 

9-913365 

147 

9-845227 

448 

10-154773 

60 

1 

708772 

3oo 

918276 

148 

845496 

448 

1 54304 

t 

3 

758932 

3  00 

918187 

845764 

448 

154286 

3 

759132 

3oo 

918099 

148 

846033 

448 

153967 

57 

4 

7593 1 2 

3oo 

918010 

148 

846802 

448 

138698 
1 53430 

36 

.5 

759492 

3  00 

912922 

148 

846570 

447 

55 

6 

759672 

299 

912888 

148 

846889 

447 

i58i6i 

54 

I 

759802 

299 

912744 

148 

847108 

447 

132892 

53 

76003 1 

299 

912655 

148 

847876 

447 

152624 

52 

9 

7602 1 1 

299 

912366 

148 

847644 

447 

152356 

5i 

10 

760890 

299 

912477 

148 

847913 

447 

152087 

5o 

II 

9-760569 

298 

9-912388 

148 

9-848181 

447 

io-i5i8i9 

49 

12 

760740 

298 

912299 

149 

848449 

4.7 

i5i55i 

48 

i3 

760927 

293 

912210 

149 

848717 

447 

i5i283 

47 

14 

761106 

293 

912121 

149 

848986 

447 

i5ioi4 

46 

i5 

761285 

298 

912081 

149 

849254 

447 

130746 

45 

i6 

761464 

298 

911942 

149 

849522 

447 

1 50478 

44 

n 

761642 

297 

911853 

149 

849790 
85oo37 

446 

l5o2IO 

43 

i8 

761821 

297 

911768 

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446 

149943 

42 

•9 

761999 

297 

91 1674 

149 

85o323 

446 

149O75 

41 

20 

762177 

297 

9ii584 

149 

830393 

446 

149407 

40 

21 

9-762356 

297 

9-9"495 

I4£^ 

9-830861 

446 

10-140189 

140871 

39 

22 

762534 

296 

9ii4o5 

147 

851129 

446 

38 

23 

762712 

296 

911813 

130 

831396 

446 

148604 

37 

24 

762889 

296 

911226 

130 

85 1664 

446 

148836 

86 

23 

763067 

296 

911186 

130 

851981 

446 

148069 

35 

26 

768245 

296 

911046 

130 

852199 

446 

147801 

34 

2? 

768422 

296 

010956 

i5o 

832466 

446 

147534 

33 

28 

768600 

293 

910866 

i5o 

852788 

443 

147267 

32 

29 

768777 

293 

910776 

i5o 

858001 

443 

146999 

3i 

3o 

768954 

293 

910686 

i5o 

858268 

445 

146782 

3o 

3i 

9-764131 

295 

9-910596 

i5o 

9-858535 

445 

10-146463 

?2 

32 

764808 

293 

910306 

130 

858802 

445 

146198 
1459 J I 

33 

764483 

294 

910415 

130 

834069 

445 

27 

34 

764662 

294 

910823 

i5i 

854886 

445 

145664 

26 

35 

764888 

294 

910235 

131 

854608 

445 

145397 
i45i3o 

25 

36 

765oi5 

294 

910144 

i5i 

854870 

445 

24 

37 

765191 

294 

910054 

i5i 

855i37 

445 

144863 

23 

38 

765367 

294 

909968 

i5i 

855404 

445 

144396 

22 

39 

765544 

293 

909S78 

i5i 

855671 

444 

144829 

21 

40 

763720 

293 

909782 

131 

855988 

444 

144062 

20 

41 

9-765896 

293 

9-909691 

i5i 

9-856204 

444 

10-143796 

19 

42 

766072 

298 

909601 

i5i 

856471 

444 

148529 
148268  ^ 

18 

43 

766247 

298 

909310 

i5i 

856787 

444 

17 

44 

766423 

298 

909419 

i5i 

837004 

444 

142996  ' 
142780 

16 

45 

766598 

292 

909828 

l52 

857270 

444 

i5 

46 

766774 

292 

909287 

132 

8573:'7 

444 

142468 

14 

47 

766949 

292 

909146 

l52 

85780J 

444 

142197 
1419^1 

i3 

48 

767124 

292 

909055 
908964 

l52 

858069 

444 

12 

49 

767800 

292 

l52 

858886 

444 

141664 

11 

5o 

767475 

291 

908878 

l52 

858602 

443 

141898 

10 

5i 

9.767649 

291 

9-908781 

l52 

9-858863 

443 

10-141182 

% 

52 

767824 

291 

908690 

l52 

859134 

443 

140866 

53 

767999 
768173 

291 

908399 

l52 

859400 

443 

140600 

1 

54 

291 

908507 

l52 

859666 

443 

140334 

6 

55 

768843 

290 

908416 

1 53 

839932 

443 

140068 

5 

56 

768522 

290 

908824 

1 53 

860198 

448 

189802 

4 

57 

768697 

290 

908233 

i53 

860464 

448 

189336 

3 

58 

768871 

290 

908141 

1 53 

860780 

443 

189270 

3 

59 

769045 

290 

908049 

1 53 

860995 

443 

189005 

1 

60 

769219 

290 

907938 

i53 

861261 

443 

188789 

0 

r 

Cosine. 

1  ^• 

j    Sine. 

1   ^' 

Cotang. 

D. 

j   Tang. 

F 

12. 

5° 

64°  I 

■-__ 

LOGARITHMIC  SINES, 

TANGENTS,  ETC 

).    Table  II. 

36° 

143° 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

f 

0 

9-769219 

290 

289 

9-907958 

i53 

9-861261 

443 

10.138739 
138473 

60 

I 

769393 

907866 

i53 

861527 

443 

59 
58 

3 

769566 

289 

907774 

1 53 

861792 

442 

i382o8 

3 

769740 

289 

907682 

1 53 

862038 

442 

137942 

57 

4 

769913 

289 

907390 

1 53 

862323 

442 

137677 

56 

5 

770087 

289 

907498 

i53 

862389 

442 

137411 

55 

6 

770260 

288 

907406 

1 53 

862854 

442 

137146 

54 

7 

770433 

288 

907314 

134 

863119 

442 

1 36881 

53 

8 

770606 

288 

907222 

1 54 

863385 

442 

i366i5 

52 

9 

770779 

288 

907129 

1 54 

863650 

442 

i3633o 

5i 

10 

770952 

288 

907037 

144 

863915 

442 

i36o85 

5o 

II 

9  771125 

283 

9-9060J5 
906832 

1 54 

9-864180 

442 

io-i3582o 

4r 

12 

771298 

287 

1 54 

864445 

442 

i35555 

4S 

i3 

771470 

''^7 

906760 

134 

864710 

442 

135290 

47 

i4 

771643 

287 

906667 

154 

864975 

441 

i35o25 

46 

i5 

771815 

287 

906575 

134 

865240 

441 

134760 

45 

i6 

771987 

287 

906482 

i54 

8655o5 

441 

134495 

44 

17 

772139 

287 

906389 

1 55 

863770 

441 

i3423o 

43 

i8 

772331 

286 

906296 

i55 

866035 

441 

133965  ■ 

42 

•9 

7725o3 

286 

906204 

i55 

866300 

441 

133700 

4! 

20 

772675 

286 

9061 11 

i55 

866564 

441 

133436 

40 

21 

9-772847 

286 

9-906018 

*53 

9-866829 

441 

io-i33i7i 

39 

22 

773018 

2S6 

903925 

i55 

867004 

441 

132906 

38 

23 

773190 

286 

903832 

i55 

867358 

441 

132642 

37 

24 

773361 

285 

905739 

i55 

867623 

441 

132377 

36 

25 

773533 

285 

905645 

i55 

867887 

441 

i32ii3 

35 

26 

773704 

285 

905552 

i55 

8681 52 

440 

1 3 1848 

34 

27 

773875 

285 

905439 

155 

868416 

440 

i3i584 

33 

28 

774046 

285 

903366 

1 56 

868680 

440 

i3i32o 

32 

29 

774217 

285 

905272 

1 56 

868945 

440 

i3io55 

3i 

3o 

774388 

284 

905179 

1 56 

869209 

440 

130791 

3o 

3i 

9-774553 

284 

9 -905085 

i56 

9-869473 

440 

io-i3o527 

29 

32 

774729 

284 

904992 

136 

869737 

440 

i3o263 

28 

33 

774899 

284 

904898 

136 

870001 

440 

1 59999 

27 

34 

775070 

284 

904804 

136 

870265 

440 

129733 

26 

35 

770240 

284 

904711 

1 56 

870529 

440 

129471 

25 

36 

775410 

283 

904617 

1 56 

870793 

440 

129207 

24 

37 

775580 

283 

904523 

1 56 

871057 

440 

128943 

23 

38 

775750 

283 

904429 

1 57 

871321 

440 

128679 
128415 

22 

39 

775920 

283 

904335 

1 57 

871585 

440 

21 

40 

776090 

283 

904241 

1 57 

871849 

439 

I28i5i 

20 

4i 

9-776259 

283 

9-904147 

1 57 

9-872112 

439 

10-1278S8 

:? 

42 

776420 

282 

904053 

157 

872376 

439 

127624 

43 

.  776598 

282 

903959 

•57 

872640 

439 

127360 

17 

44 

776768 

282 

903864 

1 57 

872903 

439 

127097 
126833 

16 

45 

776937 

282 

903770 

157 

873167 

439 

i5 

46 

777106 

282 

903676 

1 57 

873430 

439 

126570 

14 

47 

777275 

281 

903 58 I 

167 

873694 
873957 

439 

i263o6 

i3 

48 

777444 

281 

903487 

i57 

439 

1 26043 

12 

49 

777613 

281 

903392 

138 

874220 

439 

123780 

11 

5o 

777781 

281 

903298 

i58 

874484 

439 

1235l6 

10 

5i 

9-777950 

281 

9-9o32o3 

i58 

9-874747 

439 

10-125253 

t 

52 

778119 

281 

903 1 08 

1 58 

873010 

439 

1 2  f;990 

53 

778287 

280 

9o3oi4 

1 58 

875273 

438 

124727 

7 

54 

778455 

280 

902919 

i58 

875537 

438 

124463 

6 

55 

778624 

2S0 

902S24 

i58 

875800 

438 

124200 

5 

56 

778792 

280 

902729 

i58 

876063 

438 

123937 

4 

57 

778960 

280 

902634 

138 

876326 

438 

123674 

3 

58 

779128 

280 

902539 

139 

876589 

438 

12341 1 

2 

59 

779295 

279 

902444 

1 59 

876852 

438 

i23i48 

I 

60 

779463 

279 

902349 

159 

877114 

438 

122886 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

B3° 

1-2P 

0 

Table  II.   LOGARITHMIC  SINES, 

TANGENTS,  ETC. 

-^ 

87° 

/ 

142<»  1 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

r 

0 

9-779463 

279 

9.902349 
902253 

159 

9.8771 14 

438 

10-122886 

60 

I 

779631 

279 

1 59 

877377 

438 

122623 

39 
5.4 

2 

779798 

279 

902 1 58 

1 59 

877640 

438 

122360 

3 

779966 

279 

902063 

159 

l]ir& 

438 

122097 

121833 

57 

4 

780133 

279 

901067 
901872 

1 59 

438 

56 

5 

780300 

'^ 

.59 

878428 

438 

121572 

55 

6 

780467 

278 

901776 

1 39 

878691 

438 

i2i3o9 

54 

I 

780634 

'7^ 

901681 

ID9 

878933 

437 

121047 

53 

780801 

278 

901 585 

1 39 

879216 

437 

120784 

32 

9 

780968 

278 

901490 

159 

879478 

437 

120522 

5i 

10 

781 i34 

278 

901394 

160 

879741 

437 

120259 

5o 

II 

9-78i3oi 

277 

9-901298 

160 

9 -880003 

437 

10-119997 

119733 

49 

12 

781468 

277 

901202 

160 

880205 

437 

48 

i3 

781634 

277 

901 to6 

160 

880528 

437 

119472 

47 

i4 

781800 

277 

901010 

160 

880790 

437 

110210 

118948 

46 

i5 

781966 

277 

900914 

160 

88io52 

437 

43 

i6 

782132 

277 

900818 

160 

88i3i4 

437 

118686 

44 

\l 

782298 

276 

900722 

160 

881577 

437 

118423 

43 

782464 

276 

900626 

160 

881839 

437 

118161 

42 

>9 

782630 

276 

900529 

160 

882101 

437 

117899 
117637 

41 

20 

782796 

276 

900433 

161 

882363 

436 

40 

21 

9-782961 

276 

9-900337 

161 

9-882625 

436 

10-117375 

39 

22 

783127 

276 

900240 

161 

882887 

436 

117113 

38 

23 

783292 

275 

900144 

161 

883 148 

436 

116852 

3- 

24 

733438 

275 

900047 

161 

883410 

436 

1 16590 

36 

25 

783623 

275 

8999?! 

161 

8836^72 

436 

116328 

35 

26 

783788 

275 

899854 

161 

883934 

436 

1 16066 

34 

27 

783953 

275 

899757 

161 

884196 

436 

ii58o4 

33 

28 

7841 18 

275 

899660 

161 

884457 

436 

115543 

32 

29 

784282 

274 

899364 

161 

884719 

436 

115281 

3i 

3o 

7^4447 

274 

899467 

162 

884980 

436 

113020 

3o 

3i 

9-784612 

274 

9-899370 

162 

9-885242 

436 

10-114758 

29 

32 

784776 

274 

899273 

162 

885304 

436 

114496 

28 

33 

784941 

274 

899176 

162 

885765 

436 

114235 

27 

34 

785io5 

274 

899078 

162 

886026 

436 

113974 

26 

35 

785269 

273 

898981 

162 

886288 

436 

113712 

23 

36 

785433 

273 

898884 

162 

886549 

435 

11 345 1 

24 

37 

785597 

273 

89S787 

162 

886811 

435 

113189 

23 

38 

785761 

273 

898689 

162 

887072 

435 

112928 

22 

39 

785925 

273 

898592 

162 

887333 

435 

112667 

21 

40 

786089 

273 

898494 

i63 

887594 

435 

1 1 2406 

20 

41 

9-786252 

272 

9-898397 

1 63 

9-887855 

435 

10-112145 

10 

42 

786416 

272 

898299 

1 63 

888116 

435 

111884 

18 

43 

786579 

272 

898202 

163 

888378 

435 

111622 

17 

44 

786742 

272 

898104 

1 63 

888639 

435 

iii36i 

16 

45 

786906 

272 

898006 

1 63 

888900 

435 

1 1 1 1 00 

i5 

46 

787069 

272 

897908 

i63 

889 161 

435 

1 10839 

14 

47 

787232 

271 

897810 

1 63 

889421 

435 

110379 

i3 

48 

787395 

271 

897712 

i63 

889682 

435 

iio3i8 

12 

49 

787557 

271 

897614 

163 

889943 

435 

110057 

11 

5o 

787720 

271 

897516 

163 

890204 

434 

109796 

10 

5i 

9-787883 

271 

9-897418 

164 

9  -  890465 

434 

10-109535 

0 

52 

788045 

271 

897320 

164 

890725 

434 

109275 

0 

53 

788208 

271 

897222 

164 

890986 

434 

109014 
108753 

7 

54 

788370 

270 

897123 

164 

891247 

434 

6 

55 

788532 

270 

897025 

164 

891507 

434 

108493 

5 

56 

788694 

270 

896926 

164 

891768 

434 

108232 

4 

u 

788856 

270 

896828 

164 

892028 

434 

107972 

3 

789018 

270 

896729 

164 

892289 

434 

107711 

2 

59 

789180 

270 

896631 

164 

892549 

434 

107451 

I 

60 

789342 

269 

896532 

164 

892810 

434 

107190 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

12 

73 

52^ 

u 


66 

LOGARITHMIC  SINES, 

TANGENTS,  ETC 

Table  II. 

380 

1410 

1 

Sre. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

o 

9.789342 

269 

9-896532 

164 

9-892810 

434 

10-107190 
106930 

60 

I 

789504 

269 

896433 

i65 

893070 

434 

ll 

a 

789665 

269 

896335 

i65 

893331 

434 

1 06669 

3 

789827 

269 

896236 

i65 

893591 

434 

106409 

57 

4 

789988 

269 

806137 
896038 

i65 

893351 

434 

106149 

56 

5 

790149 

269 
268 

i65 

894111 

434 

■105889 

55 

6 

7903 10 

895939 

i65 

■   894372 

434 

105628 

54 

7 

790471 

268 

895840 

i65 

89^632 

433 

105368 

53 

8 

790632 

268 

895741 

i65 

894892 

433 

io5io8 

52 

9 

790793 

268 

895641 

i65 

893132 

433 

104848 

5i 

10 

790954 

268 

895542 

i65 

893412 

433 

104588 

5o 

II 

9.791115 

268 

9.895443 

166 

9.893672 

433 

10-104328 

t 

12 

791275 

267 

895343 

166 

893932 

433 

104068 

i3 

791436 

267 

893244 

166 

896192 

433 

io38o8 

47 

i4 

791596 

267 

895145 

166 

896452 

433 

103548 

46 

i5 

79'757 

267 

895045 

166 

896712 

433 

103288 

45 

i6 

791917 

267 

894945 

166 

896971 

433 

io3o29 

44 

n 

792077 

267 

894846 

166 

897231 

433 

102769 

43 

i8 

792237 

266 

894746 

166 

897491 

433 

102509 

42 

'9 

792397 

266 

894646 

166 

897731 

433 

102249 

41 

20 

792557 

266 

894546 

166 

898010 

433 

101990 

40 

21 

9-792716  • 

266 

9-894446 

167 

9.898270 

433 

10-101730 

39 

22 

792876 

266 

894346 

167 

898530 

433 

101470 

38 

23 

793o35 

266 

894246 

167 

898789 

433 

101211  !  37  1 

24 

793195 

265 

894146 

167 

899049 

432 

100951 

36 

25 

793354 

265 

894046 

167 

899308 

432 

100692 

35 

26 

793514 

265 

893946 

167 

899568 

432 

100432 

34 

27 

.793673 

265 

893846 

167 

899827 

432 

100173 

33 

28 

793832 

265 

893745 

167 

900087 

432 

099913 

32 

29 

793991 

265 

893645 

167 

900346 

432 

099654 

3i 

3o 

794130 

264 

893544 

167 

900605 

432 

099395 

3o 

3i 

9 -794308 

264 

9-893444 

168 

9.900864 

432 

10-099136 

20 

32 

794467 

264 

893343 

168 

901124 

432 

098876 

26 

33 

794626 

264 

893243 

168 

90i383 

432 

098617 

27 

34 

79475^4 

264 

893142 

168 

901642 

432 

093358 

26 

35 

794942 

264 

893041 

168 

901901 

432 

098099 

25 

36 

795101 

264 

892940 

168 

902160 

432 

097840 

24 

37 

795239 

263 

892839 

168 

902420 

432 

097580 

23 

38 

795417 

263 

892739 

168 

902679 

432 

097321 

22 

39 

795575 

263 

892638 

168 

902938 

432 

097062 

21 

40 

795733 

263 

892536 

168 

903197 

43 1 

096803 

20 

4i 

9-795891 

263 

9.892435 

169 

9.903436 

43 1 

10-096544 

19 

42 

796049 

263 

892334 

169 

903714 

43 1 

096286 

18 

43 

796206 

263 

892233 

169 

903973 

43 1 

096027 

17 

44 

796364 

262 

.  892132 

169 

904232 

43 1 

095768 

16 

45 

796521 

262 

892030 

.69 

904491 

43 1 

095509 

i5 

46 

796679 

262 

891929 

169 

904750 

43 1 

093250 

14 

47 

796836 

262 

891827 

169 

9o5oo8 

43 1 

094992 

i3 

48 

796993 

262 

891726 

169 

905267 

43 1 

094733 

12 

49 

797130 

261 

891624 

169 

905526 

43 1 

094474 

11 

5o 

797307 

261 

891523 

170 

905785 

43 1 

094215 

10 

5i 

9-797464 

261 

9-891421 

170 

9.906043 

43 1 

10-093957 

0 

52 

797621 

261 

•  891319 

170 

906302 

43 1 

093698 

8 

53 

797777 

261 

891217 

170 

906360 

43i 

093440 

7 

54 

797934 

261 

89II15 

170 

906819 

43 1 

093181 

6 

55 

798091 

261 

891013 

170 

907077 

43 1 

092923 

6 

56 

798247 

261 

890911 

170 

907336 

43 1 

092664 

4 

57 

798403 

260 

890809 

170 

907394 

43 1 

092406 

3 

53 

798560 

260 

890707 

170 

907853 

43i 

092147 

2 

59 

798716 

260 

890605 

170 

9081 11 

43o 

091889 

I 

6o 

798872 

260 

890503 

170 

90S369 

43o 

091631 

0 

1 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

Til 

° 

61"  1 

Table  II.   LOGARITHMIC  SIXES, 

TANGENl 

S,  ETC. 

A 

89° 

140«  1 

, 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9.798872 

260 

9.890503 

170 

9-908369 

43o 

io-09i63i 

60 

I 

799028 

260 

890400 

171 

908628 

43o 

091372 

s 

2 

799184 

260 

890298 

17' 

908886 

43o 

091114 

3 

799339 

259 

890195 

I7« 

909144 

43o 

090836 

57 

4 

799493 

239 

800093 

171 

909402 

43o 

090598 

56 

5 

7996D1 

239 

889990 
889888 

17' 

909660 

43o 

090340 

55 

6 

799806 

259 

171 

909918 

43o 

090082 
089823 

54 

I 

799962 
8001 17 

239 

889785 

171 

910177 

43o 

53 

259 

889682 

I7« 

910435 

43o 

089565 

52 

9 

800272 

258 

889579 

171 

910693 

43o 

089307 

5i 

10 

800427 

258 

889477 

171 

910931 

43o 

089049 

5o 

II 

9 -800582 

258 

9-889374 

172 

9-911209 

43o 

10-088791 
088533 

40 

12 

800737 

258 

889271 

172 

911467 

^3o 

48 

i3 

800892 

258 

889168 

172 

911725 

43o 

088275 

47 

i4 

801047 

258 

889064 

172 

911982 

43o 

088018 

46 

i5 

801201 

258 

888961 
888858 

172 

912240 

43o 

087760 

45 

i6 

8oi356 

257 

172 

912498 

43o 

087502 

44 

17 

8oi5ii 

257 

888755 

172 

912736 

43o 

087244 

43 

i8 

801 665 

237 

88865 1 

172 

9i3oi4 

429 

086986 

42 

19 

801819 

237 

888548 

172 

913271 

429 

086729 

41 

20 

801973 

237 

888444 

173 

913529 

429 

08647  • 

40 

21 

9'8o2i28 

257 

9-888341 

173 

9-913787 

429 

10-086213 

39 

22 

802282 

256 

888237 

173 

914044 

429 

085956 

38 

23 

802436 

256 

888 1 34 

173 

914302 

429 

085698 

37 

24 

802589 
802743 

256 

888o3o 

173 

914560 

429 

085440 

36 

23 

256 

887926 
887822 

173 

914817 

429 

o85i83 

35 

26 

802897 

256 

173 

915075 

429 

084925 

34 

27 

8o3o5o 

256 

887718 

173 

915332 

429 

084668 

33 

28 

8o3204 

236 

887614 

173 

915590 

429 

084410 

32 

29 

803357 

255 

887510 

173 

913847 

429 

084153 

3i 

3o 

8o35n 

255 

887406 

174 

916104 

429 

083896 

3o 

3i 

9-803664 

255 

9-887302 

174 

9-916362 

429 

10-083638 

?§ 

32 

8o38i7 

233 

887198 

174 

916619 

429 

08338 1 

33 

803970 

233 

887003 
886989 
886885 

174 

916877 

429 

o83i23 

27 

34 

804 ^23 

233 

174 

917134 

429 

082866 

26 

35 

804276 

234 

174 

917391 

429 

082609 

25 

36 

804428 

234 

886780 

174 

917648 

429 

082352 

24 

37 

80458 J 

234 

886676 

174 

917906 

429 

082094 

23 

38 

804734 

2  34 

886571 

•74 

918163 

428 

081837 

22 

39 

804886 

234 

886466 

174 

918420 

428 

o8i58o 

21 

40 

8o5o39 

234 

886362 

175 

9i«677 

428 

o8i323 

20 

41 

9-8o5igi 

254 

9-886257 

'■'^ 

9-918934 

423 

10-081066 

10 

42 

805343 

253 

886i52 

175 

919191 

428 

080809 

18 

43 

805495 

253 

886047 

"^\ 

919448 

428 

o8o552 

17 

44 

8o5647 

253 

885942 

175 

919705 

428 

080295 

16 

45 

8057^9 

253 

885837 

173 

919962 

428 

o8oo38 

i5 

46 

805931 

253 

885732 

175 

920219 

428 

079781 

14 

47 

806 I o3 

253 

885627 

175 

920476 

428 

079524 

i3 

48 

806254 

233 

885522 

173 

920733 

428 

079267 

12 

49 

806406 

232 

883416 

175 

920990 

428 

079010 

II 

5o 

806557 

232 

8853  m 

176 

921247 

423 

078753 

10 

5i 

9-806709 

252 

9-885205 

176 

9-92i5o3 

428 

10-078497 

0 

52 

806860 

232 

885100 

176 

921760 

428 

078240 

8 

53 

807011 

232 

884994 

176 

922017 

428 

077983 

7 

54 

807163 

232 

884889 
884783 

176 

922274 

428 

077726 

6 

55 

807314 

252 

176 

922530 

428 

077470 

5 

56 

807465 

25l 

884677 

176 

922787 

428 

077213 

4 

U 

807615 

25l 

884572 

1-.76 

923o44 

428 

076956 

3 

807766 

25l 

884466 

176 

923300 

428 

076700 

3 

59 

807917 

231 

8&t36o 

176 

923557 

427 

076443 

I 

60 

808067 

25l 

884254 

177 

923814 

427 

076186 

0 

'     Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

12s 

0 

[ 

W 

58 

LOGARITHMIC  SINES, 

TANGEXTS.  ETC 

Table 

in 

40O 

\Z9°  1 

1 

Sine. 

D. 

Cosine.   ] 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

0-808067 

25l 

9.884254 

177 

9-923814 

427 

10-076186 

60 

I 

808218 

25l 

884148 

177 

924070 

427 

075930 

^9 

2 

8o8368 

25l 

884042 

«77 

924327 

427 

075673 

58 

3 

8o85i9 

25o 

883936 

"77 

924533 

427 

075417 

57 

4 

808669 

25o 

883829 

•77 

924840 

427 

075160 

56 

5 

808819 

25o 

883723 

•77 

925096 

427 

074904 

55 

6 

808969 

25o 

883617 

177 

925352 

427 

074643 

54 

7 

8091 19 

230 

883510 

>77 

923609 

427 

074391 

53 

8 

809269 

200 

883404 

>77 

923865 

427 

074135 

52 

9 

809419 

249 

883297 

17S 

926122 

427 

073878 

5i 

10 

809569 

249 

883191 

.78 

926378 

427 

073622 

5o 

II 

9-809-18 

249 

9-883o84 

J78 

9-926634 

427 

10-073366 

% 

12 

809868 

24gr 

882977 

^'l 

926890 

427 

073110 

i3 

810017 

249 

882871 

178 

9^7147 

427 

072853 

47 

14 

810167 
8io3i6 

249 

882764 

178 

927403 

427 

072597 

46 

i5 

248 

882657 

178 

927659 

427 

072341 

43 

i6 

810465 

248 

882550 

178 

927915 

427 

072083 

44 

17 

810614 

248 

882443 

178 

928171 

427. 

071829 

43 

i8 

810763 

248 

882336 

179 

928427 

427 

071573 

42 

'9 

810912 

243 

882229 

179 

928684 

427 

071316 

41 

20 

811061 

243 

882121 

179 

928940 

427 

071060 

40 

21 

9-811210 

248 

9-882014 

179 

9-929196 

427 

10-070804 

^9 

2  2 

8ii358 

247 

881907 

•79 

929432 

427 

070548 

38 

23 

811 5o7 

247 

881799 

179 

929708 

427 

070292 

37 

24 

8ii655 

247 

881692 

179 

029964 

426 

070036 

36 

25 

811804 

247 

88 1 584 

179 

930220 

426 

069780 

35 

26 

811952 

247 

881477 

179 

930475 

426 

069525 

34 

27 

812100 

247 

881369 

J  79 

930731 

426 

069269 

33 

28 

812243 

247 

881261 

I  So 

930987 

426 

069013 

32 

29 

812396 

246 

88ii53 

180 

931243 

426 

068757 

3i 

3o 

812044 

246 

881046 

180 

931499 

426 

o685oi 

3o 

3i 

9-812692 

246 

9 -880938 
88o83o 

180 

9-931755 

426 

10-068245 

20 

32 

81284c 

246 

180 

932010 

426 

067990 
067734 

28 

33 

812988 

246 

880722 

180 

932266 

426 

27 

34 

8i3i35 

246 

8806 1 3 

180 

932522 

426 

067473 

26 

35 

8i3283 

246 

88o5o5 

180 

932778 

426 

067222 

25 

36 

8i343o 

245 

880397 

180 

933o33 

426 

066967 

24 

37 

8i3578 

245 

880289 

i8i 

933289 

426 

0667 1 1 

23 

38 

813725 

245 

880180 

181 

933543 

426 

066455 

22 

39 

81387a 

245 

880072 

181 

933800 

426 

066200 

21 

40 

814019 

245 

879963 

181 

934056 

426 

065944 

20 

41 

o-8i4i66 

245 

9-879855 

181 

9-934311 

426 

10-065689 

10 

42 

8i43i3 

245 

879746 

181 

934367 

426 

065433 

18 

43 

814460 

244 

879637 

181 

934822 

426 

065178 

17 

44 

814607 

244 

879329 

181 

935078 

426 

064922 

16 

45 

814753 

244 

879420 

181 

935333 

426 

064667 

13 

46 

814900 

244 

879311 

181 

935589 

426 

064411 

14 

47 

81 5046 

244 

879202 

182 

935844 

426 

0641 56 

i3 

48 

8i5i93 

244 

879093 

1B2 

936100 

426 

063900 

12 

49 

815339 

244 

8780S4 

182 

936355 

426 

063643 

11 

5o 

81 5485 

243 

878875 

182 

936611 

426 

063389 

10 

5i 

9-8i563i 

243 

9-878766 

182 

9-936866 

425 

io-o63i34 

0 

52 

815778 

243 

878656 

182 

937121 

425 

062879 

8 

53 

80924 

243 

878547 

182 

937377 

425 

062623 

I 

54 

816069 

243 

878438 

182 

937632 

425 

062368 

55 

816215 

243 

878328 

1S2 

937887 
93B142 

425 

0621 i3 

5 

56 

8i636i 

243 

878219 

1 83 

425 

061833 

4 

57 

8i65o7 

242 

878109 

1 83 

938398 

423 

061602 

3 

58 

8 1 6652 

242 

877999 

1 83 

938653 

423 

061347 

9 

59 

816798 

242 

877890 

1 83 

938908 

425 

061092 

I 

60 

816943 

242 

877780 

i83 

939163 

425 

060837 

a 

1 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 
49° 

Is 

J° 

Table  II.   LOGARITIOIIC  SINES, 

TANGENTS,  ETC. 

59] 

41= 

138<»  1 

/ 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9-8i69i3 

242 

9.877780 

1 83 

9-939163 

425 

10-060837 

60 

I 

8170S8 

242 

877670 

1 83 

939418 

425 

060382 

% 

2 

817233 

242 

877560 

183 

939673 

425 

060327 

3 

817379 

242 

877450  . 

183 

939928 

425 

060072 

57 

4 

817524 

241 

877340 

i83 

940183 

425 

039817 

56 

5 

817668 

241 

877230 

184 

940439 

425 

039361 

55 

6 

817813 

241 

877120 

184 

940694 

425 

059306 

54 

I 

817958 
818103 

241 

877010 

184 

940949 

425 

05905 1 

53 

241 

876899 

184 

941204 

425 

058796 

52 

9 

818247 

241 

876789 
876678 

184 

941459 

425 

058541 

5i 

10 

818392 

241 

184 

941713 

425 

058287 

5o 

II 

9-8i8536 

240 

9-876568 

184 

9-941968 

425 

10 -058032 

49 

48 

12 

818681 

240 

876457 

184 

942223 

425 

037777 

i3 

818825 

240 

876347 

184 

942478 

425 

057522 

47 

14 

818969 
819113 

240 

876236 

i85 

942733 

425 

057267 

46 

i5 

240 

876125 

1 85 

942988 

425 

057012 

45 

i6 

819257 

240 

876014 

i85 

943243 

425 

036757 

44 

<7 

819401 

240 

873904 

i85 

943498 
943762 

425 

o565o2 

43 

i8 

819545 

239 

875703 
875682 

i85 

425 

036248 

42 

•9 

819689 

239 

1 85 

944007 

425 

055993 

41 

20 

819832 

239 

875571 

i85 

944262 

425 

055738 

40 

21 

9-819976 

239 

9-875459 

i85 

9-944517 

425 

io-o55483 

S 

22 

820120 

239 

875348 

i85 

94477 1 

424 

055229 

23 

820263 

239 

875237 

1 85 

945026 

424 

034974 

37 

24 

820406 

239 

875126 

186 

945281 

424 

054719 

36 

25 

82o55o 

238 

875014 

186 

945535 

424 

034463 

35 

26 

820693 

238 

874903 

186 

945790 

.424 

o542io 

34 

27 

820836 

238 

874791 

186 

946045 

424 

053955 

33 

28 

820979 

238 

874680 

186 

946299 

424 

053701 

32 

29 

821122 

238 

874568 

186 

946554 

424 

033446 

3i 

3o 

821265 

238 

874456 

186 

946808 

424 

053192 

3o 

3i 

9-821407 

238 

9-874344 

186 

9-947063 

424 

10-052937 

29 

28 

32 

82i55o 

238 

874232 

187 

947318 

424 

052682 

33 

821693 

237 

874121 

187 

947572 

424 

052428 

27 

34 

821835 

237 

874009 

187 

947827 

424 

052173 

26 

35 

821977 

237 

873^,6 

187 

948081 

424 

051919 

25 

36 

822120 

237 

8-3-S4 

187 

948335 

424 

o5i665 

24 

37 

822262 

237 

873672 

187 

948590 

424 

o5i4io 

23 

38 

822404 

237 

873560 

187 

948844 

424 

o5ii56 

22 

39 

822546 

237 

873448 

187 

949090 
949333 

424 

050901 

21 

40 

822688 

236 

873335 

187 

424 

o5o647 

20 

41 

9-822830 

236 

9-873223 

187 

9-949608 

424 

io-o5o392 
o5oi38 

\% 

42 

822972 

236 

873110 

188 

949862 

424 

43 

823114 

236 

872998 
-   872685 

188 

950116 

424 

049884 

17 

44 

823255 

236 

188 

950371 

424 

049629 

16 

45 

823397 

236 

872772 

188 

950625 

424 

049373 

i5 

46 

823539 

236 

872659 

188 

950879 

424 

049121 

048867 

14 

8 

823680 

235 

872547 

188 

95ii33 

424 

i3 

823821 

235 

872434 

188 

95i388 

424 

048612 

13 

49 

823963 

235 

872321 

188 

951642 

424 

048358 

It 

5o 

824104 

235 

872208 

188 

951896 

424 

048104 

10 

5i 

9-824245 

235 

9-872095 

189 

9-952i5o 

4U 

10-047850 

I 

52 

824386 

235 

§71981 

189 

952405 

424 

047395 

53 

824527 

235 

871868 

189 

952659 

424 

047341 

I 

34 

824668 

234 

871755 

189 

952913 

424 

047087 

55 

824808 

234 

871641 

189 

953167 

423 

046833 

5 

56 

824949 

234 

871528 

189 

953421 

423 

046579 

4 

57 

823090 

825230 

234 

871414 

i«9 

953675 

423 

U4632D 

3 

58 

a34 

871301 

1S9 

953929 

423 

046071 

3 

59 

825371 

234 

871187 

189 

954183 

423 

045817 

I 

6o 

8255ii 

234 

871073 

190 

954437 

423 

045563 

0 

'  1   Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

1 

131 

0 

4 

60 

LOGARITHMIC  SINES, 

TANGENTS,  ETC 

Table  II. 

42° 

137* 

'  1    Sine.    1 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang.    / 

0 

9-8255ii 

234 

9-871073 

190 

9-954437 

423 

10-045563 

60 

I 

825651 

233 

870960 

190 

954691 

423 

045309 

U 

2 

825791 

233 

870846 

190 

954946 

423 

o45o54 

3 

825931 

233 

870732 

190 

955200 

423 

044800 

57 

4 

826071 

233 

870618 

190 

955454 

423 

044546 

56 

5 

826211 

233 

870504 

190 

955708 

423 

044292 
044039 

55 

6 

826351 

233 

870390 

190 

955961 

423 

54 

I 

826491 

233 

870276 

190 

936215 

423 

043785 

53 

826631 

233 

870161 

190 

936469 

423 

043531 

52 

9 

826770 

232 

870047 

191 

956723 

423 

043277 

5i 

10 

826910 

232 

869933 

191 

956977 

423 

o43o23 

5o 

II 

9-827049 

232 

9-869818 

191 

9-957231 

423 

10-042769 

0425l3 

49 

12 

827189 

232 

869704 

191 

937485 

423 

48 

i3 

827328 

232 

869589 

191 

937739 
957993 

423 

042261 

47 

14 

827467 

232 

869474 

191 

423 

042007 

46 

i5 

827606 

232 

869360 

191 

958247 

423 

041753 

45 

i6 

827745 

232 

869245 

191 

958500 

423 

o4i5oo 

44 

17 

827884 

23l 

869130 

191 

958754 

423 

041246 

43 

i8 

828023 

23l 

869015 

192 

959008 

423 

040992 

42 

'9 

828162 

23l 

868900 

192 

959262 

423 

040738 

41 

20 

828301 

23l 

868785 

192 

959516 

423 

040484 

40 

21 

9-828439 

23l 

9-868670 

192 

9-959769 

423 

io-o4o23i 

39 

38 

22 

828578 

23l 

868555 

192 

960023 

423 

039977 

23 

828716 

23  I 

868440 

192 

960277 

423 

039723 

37 

24 

828855 

23o 

868324 

192 

960530 

4?3 

039470 

36 

23 

828993 

23o 

86S209 

192 

960784 

423 

039216 

35 

26 

829131 

23o 

868093 

192 

961038 

423 

038962 

34 

11 

829269 

23o 

867978 

193 

961292 

423 

038708 

33 

829407 
829545 

23o 

867862 

193 

961545 

423 

038455 

32 

29 

23o 

867747 

193 

961799 

423 

o3820i 

3i 

3o 

829683 

23o 

867631 

193 

962052 

423 

037948 

3o 

3i 

9-829821 

229 

9-867515 

193 

9-962306 

423 

10-037694 

29 

32 

829959 

229 

867399 

193 

962560 

423 

037440 

28 

33 

83oog7 
830234 

229 

867283 

193 

962813 

423 

037187 

27 

34 

229 

867167 

193 

963067 

423 

036933 

26 

35 

83o372 

229 

867051 

193 

963320 

423 

o3668o 

25 

36 

83o5o9 

229 

866935 

194 

963574 

423 

036426 

24 

37 

83o646 

239 

866819 

194 

963828 

423 

036172 

23 

38 

830784 

229 

866703 

194 

964081 

423 

035919 

22 

39 

830921 

228 

866586 

194 

964335 

423 

035665 

21 

40 

83io58 

228 

866470 

194 

964588 

422 

o354i2 

20 

41 

9-83II95 

228 

9-866353 

194 

9.964842 

422 

io-o35i58 

\l 

42 

83i332 

228 

866237 

194 

965095 

422 

o349o5 

43 

831469 

228 

866120 

194 

965349 

422 

o3465i 

17 

44 

83 1 606 

228 

866004 

195 

965602 

422 

034398 

16 

45 

831742 

228 

865887 

195 

965855 

422 

034145 

i5 

46 

831879 

228 

865770 

195 

966109 

422 

033891 

14 

47 

832015 

227 

865653 

195 

966362 

422 

033638 

i3 

48 

832152 

227 

865536 

195 

966616 

422 

033384 

12 

49 

832288 

227 

865419 

195 

966869 

422 

o33i3i 

II 

5o 

832425 

227 

865302 

IS5 

967123 

422 

032877 

10 

5i 

9-832561 

227 

9-865i85 

195 

9-967376 

422 

10-032624 

I 

52 

832697 

227 

865o68 

195 

967629 

422 

032371 

53 

832833 

227 

864950 

195 

967883 

422 

032117 

7 

54 

832969 
833io5 

226 

864833 

196 

968136 

422 

o3i864 

6 

55 

226 

864716 

196 

968389 

422 

o3i6ii 

5 

56 

833241 

226 

864598 

196 

968643 

422 

o3i357 

4 

^I 

833377 

226 

864481 

196 

968896 

422 

o3ii(4 

3 

58 

833512 

226 

864363 

196 

969149 

422 

o3o83£ 

3 

59 

833648 

220 

864245 

196 

969403 

422 

o3o597 

I 

60 

833783 

226 

864127 

I<v6 

969656 

422 

o3o344 

0 

f 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang.   1  / 

"131 

)0 

47° 

Table  II.   LOGARITHMIC  SINES, 

TANGENTS,  ETC. 

61 

43° 

Sine. 

138''  1 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9-833783 

226 

9.864127 

196 

9-969656 

422 

10 -030344 

60 

1 

833919 

225 

864010 

196 

969909 

422 

030091 
029838 

u 

2 

834o54 

225 

863892 

197 

970162 

422 

3 

834189 

225 

863774 

197 

970416 

422 

029384 

57 

4 

834323 

225 

863656 

»97 

970669 

422 

029331 

56 

5 

834460 

225 

863538 

•97 

970922 

422 

029078 

55 

6 

834595 

225 

863410 

•97 

971175 

422 

028825 

54 

I 

834730 

223 

863301 

•97 

971429 

422 

028571 

53 

834865 

225 

863 1 83 

•97 

971682 

422 

0283i8 

52 

9 

834999 

224 

863o64 

•97 

971935 

422 

028065 

5i 

10 

835i34 

224 

862946 

198 

972188 

422 

027812 

5o 

II 

9 -835269 

224 

9-862827 

198 

9-972441 

422 

10-027559 

-49 

12 

8354o3 

224 

862709 

198 

972695 

422 

027303 

48 

i3 

835538 

224 

862590 

198 

972948 

422 

027052 

47 

14 

835672 

224 

862471 

198 

973201 

422 

026799 

46 

i5 

835807 

224 

862353 

198 

973434 

422 

026546 

45 

i6 

835941 

224 

862234 

.98 

973707 

422 

026293 

44 

'7 

836075 

223 

862ii5 

198 

973960 

422 

026040 

43 

i8 

836209 

223 

861996 

198 

974213 

422 

025787 

42 

19 

836343 

223 

861877 

198 

974466 

422 

025534 

41 

20 

836477 

223 

861758 

•99 

974720 

422 

025280 

40 

21 

9-836611 

223 

9-86i638 

•99 

9-974973 

422 

10-025027 

!? 

22 

836745 

223 

86i5i9 

199 

975226 

422 

024774 

23 

836878 

223 

861400 

•99 

975479 

422 

024321 

37 

24 

837012 

222 

861280 

•99 

975732 

422 

024268 

36 

25 

837146 

222 

861161 

•99 

975985 

422 

024013 

35 

26 

837279 

222 

861041 

•99 

976238 

422 

023762 

34 

27 

837412 

222 

860922 

•99 

976491 

422 

023309 

33 

28 

837546 

222 

860802 

•99 

976744 

422 

023256 

32 

29 

837679 

222 

860682 

200 

976997 

422 

o23oo3 

3i 

3o 

837812 

222 

86o562 

200 

977250 

422 

022750 

So 

3i 

9-837945 

222 

9-860442 

200 

9-9775o3 

422 

10-022497 

20 

32 

838078 

221 

86o322 

200 

977736 

422 

022244 

28 

33 

8382 II 

221 

860202 

200 

978009 

422 

021991 

27 

34. 

838344 

221 

860082 

200 

978262 

422 

021738 

26 

35 

838477 

221 

859962 

200 

97851 5 

422 

021485 

25 

36 

838610 

221 

859842 

200 

978768 

422 

021232 

24 

3? 

838742 

221 

859721 

201 

979021 

422 

020979 

23 

38 

838875 

221 

839601 

201 

979274 

422 

020726 

22 

39 

839007 

221 

859480 

201 

979327 

422 

020473 

21 

40 

839140 

220 

839360 

201 

979780 

422 

020220 

20 

41 

9-839272 

220 

9-839239 

201 

9-980033 

422 

10-019967 

19 

42 

839404 

220 

830110 

838998 

201 

980286 

422 

019714 

43 

839536 

220 

201 

980538 

422 

019462 

•7 

44 

839668 

220 

838877 

201 

980791 

421 

019209 

16 

45 

839800 

220 

858756 

202 

981044 

421 

018956 

i5 

46 

839932 

220 

858635 

202 

981297 

421 

018703 

14 

47 

840064 

219 

858514 

202 

981530 

421 

018450 

i3 

48 

840196 

219 

858393 

202 

981803 

421 

O18197 

12 

49 

840328 

219 

858272 

202 

982056 

421 

017944 

II 

5o 

840459 

219 

858i5i 

202 

982309 

421 

OI769I 

10 

5i 

9.840591 

219 

9-858o2o 

202 

9-982562 

421 

10-017438 

9 

52 

840722 

219 

857908 

202 

982814 

421 

OI7186 

8 

53 

840854 

219 

857786 

202 

983067 

421 

016933 

7 

54 

840985 

219 

857665 

203 

983320 

421 

016680 

6 

55 

841116 

218 

857543 

2o3 

983573 

421 

016427 

5 

56 

841247 

218 

837422 

2o3 

983826 

421 

O16174 

4 

57 

841378 

218 

857300 

203 

984079 

421 

OI592I 

3 

58 

841509 

218 

857,78 

203 

984332 

421 

015668 

3 

59 

841640 

218 

857056 

2o3 

984584 

421 

oi54i6 

I 

60 

841771 

218 

856934 

2o3 

984837 

421 

oi5i63 

0 

/ 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

13S 

° 

t6° 

62 

LOGARITHMIC  SINES, 

TANGENTS,  ETC.    Table  II. 

l4° 

1350  1 

1 

Sine. 

D. 

Cosine. 

D. 

Tang. 

D. 

Cotang. 

/ 

0 

9-841771 

218 

9-856934 

2o3 

9.984837 

421 

io.oi5i63 

60 

I 

841902 

218 

856812 

2o3 

985090 

421 

OI49IO 

% 

2 

842033 

218 

856690 

204 

985343 

421 

014657 

3 

842163 

217 

856568 

204 

985596 

421 

014404 

57 

4 

842294 

217 

856446 

204 

985848 

421 

0I4I32 

56 

5 

842424 

217 

856323 

204 

986101 

421 

013899 

55 

6 

842555 

217 

856201 

204 

986354 

421 

oi3646 

54 

7 

842685 

217 

856078 

204 

986607 

421 

013393 

53 

8 

842815 

217 

855956 

204 

986860 

421 

oi3i4o 

52 

9 

842946 

217 

855833 

204 

987112 

421 

012888 

5i 

10 

843076 

217 

8557 1 1 

2o5 

987365 

421 

012635 

5o 

II 

9-843206 

216 

9-855588 

2o5 

9-987618 

421 

10-012382 

49 

12 

843336 

216 

855465 

2o5 

987871 

421 

012129 

48 

i3 

843466 

216 

855342 

205 

988123 

421 

011877 

47 

i4 

843595 

216 

855219 

2o5 

988376 

421 

01 1624 

46 

i5 

843725 

216 

855096 

2o5 

988629 

421 

.011371 

45 

i6 

843855 

216 

854973 
854850 

2o5 

988882 

42: 

011118 

44 

17 

843984 

216 

2o5 

0B9134 

421 

010866 

43 

i8 

8441 14 

2l5 

854727 

206 

989387 

421 

oio6i3 

42 

'9 

844243 

2l5 

8546o3 

206 

989640 

421 

oio36o 

41 

20 

844372 

2l5 

854480 

206 

989893 

421 

010107 

40 

21 

9-844302 

2l5 

9-854356 

206 

9-990145 

421 

10-009855 

39 

22 

844631 

2l5 

854233 

206 

990398 
990631 

421 

009602 

38 

23 

844760 

2l5 

834109 

206 

421 

009349 

37 

24 

844889 

2l5 

853986 

206 

990903 

421 

009097 

36 

23 

843018 

2l5 

853862 

206 

99 1 1 56 

421 

008844 

35 

26 

845147 

2l5 

853738 

206 

991409 

421 

008391 

34 

27 

845276 

214 

853614 

207 

991662 

421 

008338 

33 

28 

8454o5 

214 

853490 

207 

991914 

421 

008086 

32 

29 

845533 

214 

853366 

207 

992167 

421 

007833 

3i 

3o 

845662 

214 

853242 

207 

992420 

421 

007380 

3o 

3i 

9-845790 

214 

9-853ii8 

207 

3-992672 

421 

10-007328 

29 

32 

845919 

214 

832994 

207 

992925 

421 

007075 

28 

11 

846047 

214 

852869 

207 

993178 

421 

006822 

27 

34 

846175 

214 

852743 

207 

993431 

421 

006569 

.26 

35 

8463o4 

214 

852620 

207 

993683 

421 

oo63 1 7 

25 

36 

846432 

2l3 

852496 

208 

9g3g36 

421 

006064 

24 

37 

846560 

2l3 

852371 

208 

994189 

421 

oo58ii 

23 

38 

846688 

2l3 

852247 

208 

994441 

421 

005559 

22 

39 

846816 

2l3 

852122 

208 

994694 

421 

oo53o6 

21 

40 

846944 

2l3 

85 1997 

208 

994947 

421 

003053 

20 

41 

9-847071 

2l3 

9-85i872 

208 

9-993199 

421 

10-004801 

19 

42 

847199 

2l3 

851747 

208 

995452 

421 

004548 

18 

43 

847327 

2l3 

831622 

208 

993703 

421 

004295 

17 

44 

847454 

212 

85 1497 

209 

995957 

421 

004043 

16 

45 

847582 

212 

85i372 

209 

996210 

421 

003790 

i5 

46 

847709 

212 

85i246 

209 

996463 

421 

003537 

14 

% 

847836 

212 

85ii2i 

209 

9967 1 5 

421 

003285 

i3 

847964 

212 

850996 

209 

996968 

421 

oo3o32 

12 

49 

848091 

212 

850870 

209 

997221 

421 

002779 

II 

00 

848218 

212 

85o745 

209 

997473 

421 

002527 

10 

5i 

9-848343 

212 

9-85o6i9 
830493 

209 

9-997726 

421 

10-002274 

I 

32 

848472 

211 

21c 

997979 

421 

002021 

53 

848399 

211 

83o368 

210 

998231 

421 

001769 

1 

54 

848726 

211 

830242 

210 

998484 

421 

ooi5i6 

6 

55 

848832 

211 

83oii6 

210 

998737 

421 

001263 

5 

56 

848979 

211 

849990 

210 

998989 

421 

OOIOII 

4 

^2 

849106 

211 

849864 

210 

999242 

421 

000738 

3 

58 

849232 

211 

849738 

210 

999495 

421 

ooo5o5 

2 

59 

849359 

211 

84961 1 

210 

999747 

421 

000253 

I 

60 
/ 

849483 

211 

849485 

210 

10  000000 

421 

1 0-000000 

0 

Cosine. 

D. 

Sine. 

D. 

Cotang. 

D. 

Tang. 

/ 

13^ 

1° 

15° 

TABLE    III.. 


NATURAL    SINES    AND   TANGENTS) 

TO       , 

EVERY  DEGREE  AND  MINUTE  OF  THE  QUADRANT. 


If  the  given  angle  is  less  than  45°,  look  for  the  degrees  and  the  title  of  tlit 
ojlumn,  at  the  top  of  the  page ;  and  for  the  minntes  on  the  left.  But  if  the  angle 
is  between  45°  and  90°,  look  fur  the  degrees  and  the  title  of  the  column,  at  tlic 
lottom;  and  for  the  minutes  on  the  rk/ld. 

The  Secants  and  Cosecants,  wliich  are  not  inserted  in  this  table,  may  be  easily 
supplied.  If  I  be  divided  by  the  cosine  of  an  arc,  the  quotient  will  be  the  seca:\l 
of  that  arc.    And  if  i  be  divided  by  tbo  sine,  the  quotient  will  be  the  cosecant. 

The  values  of  the  Sines  and  Cosines  are  less  than  a  unit,  and  are  given  in  deci- 
mals, although  the  decimal  point  is  not  printed.  So  also,  the  tangents  of  arcs  kbc 
than  45°,  and  cotangents  of  arcs  greater  than  45°,  are  less  than  a  unit  nn  1  aio  ts- 
pessed  in  decimals  with  tl.e  deciaal  pcont  omitted. 


64            NATURAL  SIXES  AND  COSINES.      Table  III.  | 

/ 

0° 

1° 

2° 

3° 

4° 

t 

Sine. 

(/'osine. 

Sine. 

Cosine. 

Sine.  Cosine. 

Sine.  1  Cosine. 

Sine. 

Cosine. 

0 

00000 

Unit. 

01745 

99985 

03490 

99939 

05234  {  99863 

06976 

99756 

60 

I 

00020 

Unit. 

01774 

99984 

o33l9 

99938 

o5263  '  99861 

07005 

99734 

59 

2 

ooo58 

Unit. 

oi8o3 

99984 

03548  1  99937 

05292 

99S60 

07034 

99752 

58 

3 

00087 

Unit. 

oi832 

99983 

03577 

99936 

o532i 

99858 

07063 

99730 

57 

4 

001 16 

Unit. 

01862 

99983 

o36o6 

99933 

o535o 

99857 

07092 

99748 

56 

5 

00145 

Unit. 

01891  99982 

03635 

99934 

05379 

99855 

071 21 

99746 

55 

6 

00175 

Unit. 

01920  99982 

03664 

99933 

o54o8 

99854 

07l5o 

99744 

54 

7 

00204 

Unit. 

01949  99981 

03693 

99932 

05437 

99352 

07170 

99742 

53 

8 

00233 

Unit. 

i  01978  99980 

03723 

99931 

o5466 

99851 

07208 

99740 

52 

9 

00262 

Unit. 

02007  99980 

03752  99930 

03495 

99849 

07237 

99738 

5i 

10 

00291 

Unit. 

O2o36  99979 

03781  99929 

o5524 

99847 

07266 

99736 

5o 

II 

oo32o 

99999 

o2o65  99970 
02094  99978 

o38io 

99927 

o5553 

99846 

07295 

99734 

49 

12 

00349 

99999 

03839 

99926 

05582 

99844 

07324 

99731 

48 

i3 

00378 

99999 

02123  99977 

03868 

99925 

o56ii 

99842 

07353 

99729 

47 

14 

00407 

99999 

02 1 52  99977 

03897 

99924 

o564o 

99841 

07382 

99727 

46 

i5 

00436 

99999 

02181  99976 

03926 

99923 

05669 

99839 

0741I 

99725 

45 

i6 

00465 

99999 

0221 1  99976 

03955 

99922 

05698 

99838 

07440  1  99723 

44 

17 

00495 

99999 

02240  99975 

03984 

99921 

03727 

99836 

07469 

99721 

43 

i8 

oo524 

99999 

02260  99974 

o4oi3 

99919 

1 o5756 

99834 

07498 

99719 

42 

19 

oo553 

99998 

0229S !  99974 

04042 

99918 

1 05785 

99833 

07527 

99716 

41 

20 

oo582 

9999S 

02327 

99973 

04071 

999 '7 

o58i4 

9983 1 

07556 

997 '4 

40 

21 

0061 1 

99998 

1  02356 

99972 

04100 

99916 

03844 

99829 

07585 

99712 

39 

22 

00640 

99998 

02385 

99972 

04129 

99913 

05873 

99827 

07614 

99710 

38 

23 

00660 
00690 

99998 

02414 

99971 

o4i5g 

99913 

05902 

99826 

07643 

99708 

V 

24 

99998 

02443 

99970 

04188 

99912 

05931 

99824 

07672 

99705 

3o 

25 

00727 

99997 

02472 

99969 

04217 

9991 1 

05960 

99822 

07701 

99703 

35 

26 

00736 

99997 

02301 

99969 

04246 

99910 

05989 

99821 

07730 

99701 

34 

27 
28 

00785 

99997 

o253o 

99968 

04275 

99909 

06018 

99819 

07759 

90699 

33 

00814 

99997 

02360 

99967 

o43o4 

99907 

06047 

99817 

07788 

99696 

32 

29 

00844, 

99996 

02589  1  99966 

04333 

99906 

06076 

99815 

07817 

99694 

3i 

3o 

00873 

99996 

02618 

99966 

04362 

99905 

o6io5 

99813 

07846 

99692 

So 

3i 

00902 

99996 

02647 

99965 

04391 

99904 

06 1 34 

99S12 

07875 

99689 

% 

32 

00931 

99996 

02676 

99964 

04420 

99902 

06 1 63 

99810 

07904 

99687 

33 

00960 

99995 

02705 

99963 

04440 

99901 

06192 

99808 

07933 

99685 

27 

34 

00989 

99993 

02734 

99963 

04478 

99900 

06221 

99806 

07962 

99683 

26 

35 

01018 

99993 

02763 

99962 

04507 

99898 

o625o 

99804 

07991 

99680 

25 

36 

01047 

99995 

02792 

99961 

04536 

99897 

06270 

99803 

08020 

99678 

24 

37 

01076  1  99994 

02821 

99960 

04565 

99896 

o63o8 

99801 

08049 

99676 

23 

3S 

01103 

99994 

0285o 

99939 

04394 

99894 

06337 

99799 

08078 

99673 

22 

39 

oii34 

99994 

02879 

99939 

04623 

99893 

06366 

99797 

08107 

99671 

21 

40 

01164 

99993 

02908 

99938 

04653 

99892 

06395 

99795 

o8i36 

99668 

20 

41 

01193 

99993 

02938 

99957 

04682 

99800 

06424 

99793 

o8i65 

99666 

19 

42 

01222 

99993 

02967 

99936 

047 1 1 

99889 

06453 

99792 

08194 

99664 

18 

43 

OI25l 

99992 

02996 

99933 

04740 

99888 

06482 

99790 

08223 

99661 

n 

44 

01280 

99992 

o3o25 

99934 

04760 

99886 

o65ii 

99788 

08252 

99659 

16 

43 

01 309 

99991 

o3o54 

99933 

04798 

99885 

o654o 

99786 

08281 

99657 

i5 

46 

0x338 

99991 

o3o83 

99952 

04827 

99883 

06569 

99784 

o83io 

99654 

14 

47 

01367 

99991 

03ll2 

99932 

04856 

99882 

06598 

99782 

08339 

99652 

i3 

48 

01396 

99990 

o3i4i 

99931 

04885 

99881 

06627 

99780 

08368 

99649 

12 

49 

01425 

99990 

o3i70 

99930 

04914 

99879 

o6656 

99778 

08897 

99647 

II 

5o 

01454 

99989 

o3i99  99949 

04943 

99878 

06685 

99776 

08426 

99644 

10 

5i 

01483 

99989 

03228 

99948 

04972 

99876 

06714 

99774 

08455 

99642 

9 

52 

oi5i3 

99989 

o3257 

99947 

o5ooi 

99875 

06743 

99772 

08484 

99639 

8 

53 

01542 

99988 

03286 

99946 

o5o3o 

99873 

06773 
06802 

99770 

o85i3 

99637 

7 

54 

01371 

99988 

o33i6 

99943 

o5o59 

99872 

99768 

08542 

99635 

6 

55 

01600 

99987 

03345 

99944 

o5o88 

99870 

o683i 

99766 

08571 

99632 

5 

56 

01629 

99987 

03374 

99943 

o5ii7 

99869 

06860 

99764 

08600 

99630 

4 

57 

01658 

99986 

o34o3 

99942 

o5i46 

99867 

06889 

99762 

08629 

99627 

3 

58 

01687 

99986 

03432 

99941 

05175 

99866 

06918 

99760 

o8658 

99625 

3 

59 

01716 

99985 

03461 

99940 

o52o5 

99864 

06947 

99758 

08687 

99622 

I 

60 

01745 

99985 

03490 

99939 

o5234 

99863 

06976 

99706 

08716 

99619 

0 

/ 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

t 

89° 

88° 

87° 

86° 

85° 

Table  III. 

NATURAL 

SIXES  AXD  COSIXES. 

65  I 

1 

S 

0 

6° 

7° 

8° 

90 

/ 
60 

Sine 

Cosine. 

Sine.  .Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Bine. 

Cosine. 

0 

08716 

99619 

10453 

99452 

12187 

99255 

13917 

99027 

15643 

98769 

I 

08745 

99617 

10482 

99449 

12216 

99231 

13946 

99023 

13672 

98764 

5g 

2 

08774 

99614 

io5ii 

99446 

12245 

99248 

13975 

99019 

15701 

98760 

58 

3 

o88o3 

99612 

io54o 

99443 

12274 

99244 

14004 

99013 

1 5730 

98755 

57 

4 

o883i 

99609 

10569 

99440 

I2302 

99240 

i4o33 

99011 

15758 

98751 

56 

5 

08860 

99607 

10597 

99437 

I233i 

99237 

14061 

99006 

15787 

98746 

55 

6 

08889 

99604 

10626 

99434 

i236o 

99233 

14090 

99002 

i58i6 

98741 

54 

7 

08918 

99602 

io655 

99431 

12389 
I24I8 

99230 

14119 

98998 

15843 

98737 

53 

8 

08947 

99399 

10684 

99428 

99226 

14148 

98994 

15873 

98732 

52 

9 

08976 

99596 

10713 

99424 

12447 

99222 

14177 
i42o5 

98990 

15902 

98728 

5i 

10 

09005 

99394 

10742 

99421 

12476 

99219 

98986 

15931 

98723 

5o 

II 

09034 

993QI 

10771 

99418 

i25o4 

99213 

14234 

98982 

15959 

98718 

49 

12 

09063 

99388 

10800 

.99415 

12533 

992 1 1 

14263 

98978 

15988 

98714 

i^^ 

i3 

09092 

99586 

10829 

99412 

12562 

99208 

14292 

98973 

16017 

98709 

47 

i4 

09121 

99583 

io858 

99409 

12591 

99204 

14320 

98969 
98965 

16046 

98704 

46 

i5 

ogiSo 

99580 

10887 

99406 

12620 

99200 

14349 

16074 

98700 

45 

i6 

09179 

99578 

10916 

99402 

12649 
12678 

99197 

14378 

98961 

i6io3 

98695 

44 

17 

09208 

99373 

10945 

99399 

99103 

14407 

98957 

i6i32 

98690 

43 

i8 

09237 

99372 

10973 

99396 

12706 

99189 

14436 

98953 

16160 

98686 

42 

'9 

09266 

99D70 

11002 

99393 

12735 

99186 

14464 

98948 

16180 
16218 

98681 

41 

20 

09293 

99367 

iio3i 

99390 

12764 

99182 

14493 

98944 

98676 

40 

21 

09324 

99364 

11060 

99386 

12793 

99178 

14522 

98940 

16246 

98671 

39 

22 

09353 

99562 

11080 

99383 

12822 

99175 

i455i 

98936 

16275 

98667 

38 

23 

093  82 

99339 

11118 

99380 

I285i 

99171 

14380 

98931 

i63o4 

98662 

37 

24 

0941 1 

99356 

1 1147 

99377 

12880 

99167 

14608 

98927 

16333 

98657 

36 

23 

09440 

99553 

1 1 176 

99374 

12908 

99163 

14637 

98923 

i636i 

o8652 

35 

26 

09469 
09498 

99331 

11203 

99370 

12937 

99160 

14666 

98919 

16390 

98648 

34 

27 

99348 

11234 

99367 

12966 

99156 

14695 

98914 

16419 

98643 

33 

23 

09527 

99345 

11263 

99364 

12995 

99132 

14723 

98910 

16447 

98638 

32 

29 

09336 

99542 

11291 

99360 

i3o24 

99148 

14752 

98906 

16476 

98633 

3i 

3o 

09385 

99540 

Il320 

99357 

i3o53 

99144 

14781 

98902 

i65o5 

98629 

3o 

3i 

09614 

99537 

1 1 340 

99354 

i3o8i 

99'4i 

14810 

98897 

16533 

98624;  29 

32 

09642 

99334 

1 1378 j  99351 

i3iio  99137 

14838 

.98893 
98889 

16562 

986191  2S 

33 

09671 

99531 

11407  99347 

i3i39 

99133 

14867 

16591 

986141  27 

34 

09700 

99528 

1 1436  99344 

i3i68 

99129 

14896 

98884 

16620 

98609  !  26 

35 

09729 

99526 

11463  99341 

i3i97 

9912D 

14925 

98880 

16648 

98604 

23 

36 

09738 

99523 

11494  99337 

13226 

99122 

14954 

98876 

16677 

98600 

24 

37 

09787 

99520 

11 523  99334 

i3254 

99118 

14982 

9^871 

16706 

98595 

23 

38 

09816 

99517 

11 552  99331 

13283 

99114 

i5oii 

98867 

16734 

98590 

22 

39 

09845  995 1 4 

ii58o 

99327 

1 33 1 2 

99110 

i5o4o 

98863 

16763 

98585 

21 

40 

09874 

99511 

1 1 609 

69324 

i334i 

99106 

1 5069 

98858 

16792 

98580 

20 

41 

09903 

99308 

11638 

99320 

13370 

99102 

i5097 

98854 

16820 

98573 

19 

42 

09932 

995o6 

1 1667 

99317 

13399 

99098 

i5i26 

98849 

16849 

98570 

18 

43 

09961  995o3 

1 1 696 

99314 

13427 

99094 

i5i55 

98843 

16878 

98565 

"7 

44 

09990  99500 

11725 

99310 

i3456 

99091 
99087 

I3i84 

98841 

16906 

98561 

16 

45 

10019  99497 

1 1 754 

99307 

13485 

l5212 

98836 

16935 

98556 

i5 

46 

10048  99494 

11783 

993o3 

i35i4 

99083 

i524i 

98882 

16964 

9835, 

14 

47 

10077 

99491 

11812 

99300 

1 3543 

99079 

13270 

98827 

16992 

98546 

i3 

48 

10106 

99488 

11840 

99297 

13572 

99073 

13299 

98823 

17021 

9^?i' 

12 

49 

ioi35 

99485 

1 1 869 

99293 

i36oo 

99071 

15327 

98818 

17050 

98336 

II 

DO 

1 01 64 

99482 

11898 

99200 
99286 

i3629 

99067 

15356 

98814 

17078 

98531 

10 

5i 

10192 

99479 

11927 

13658 

99063 

15385 

98809 

17107 

98526 

9 

52 

10221 

99476 

11956 

99283 

13687 

99059 

i54i4 

98803 

17136 

98521 

8 

53 

I025o 

99473 

11985 

99279 

13716 

99053 

13442 

98800 

17164 

98316 

n 

54 

10279 

99470 

12014 

99276 

13744 

9905 1 

15471 

98796 

17193 

98511 

6 

55 

io3o8 

99467 

12043 

99272 

13773 

99047 

i55oo 

98701 

17222 

98306 

5 

56 

10337 

99464 

12071 

99269 

i38o2 

99043 

15529 

17250 

98501 

4 

% 

io366 

99461 

12100 

99263 

i383i 

99039 
99035 

l5337 

9878a 

17279 

98496 

3 

10395 

99458 

12129 
I2I58 

99262 

i386o 

15586 

98778 

17308 

98491 

a 

59 

10424 

99433 

99258 

13889 

9903 1 

i56i5 

98773 

17336 

98486 

I 

6o 

10453  \  99452  1 

12187 

99255 

13917 

99027 

15643 

98769 

17365 

98481 

0 

/ 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sin«. 

Cosine.  Sine. 

/ 

84° 

83° 

se^" 

81  = 

80= 

66 

XATURAL  SIXES  AND  COSINES.      Tadle  III. 

1 

10° 

11° 

12° 

13° 

14° 

t 

Sine.  Cosine. 

Sine. 

Cosine. 

Sine.  Cosine. 
20791  197813 

Sine. 

Cosine. 

Sine. 

Cosine. 

0 

17365 

98481 

19081 

98 1 63 

22495 

97437 

24192  97o3o 

60 

I 

17393 

98476 

1910Q 

98:57 

20820 

97809  1 

22523 

97430 

24220 

97oi23 

59 

58 

2 

17422 

9'^47i 

19138 

98152 

20848 

97803  1 

22552 

97424 

24249 

97015 

3 

17431 

98466 

19167 

98146 

20877 

97797 

22  58o 

97417 

24277 

97008 

57 

4 

17479 

98461 

19193 

98140 

20905 

9779' 

22608 

97411 

243o5 

97001 

56 

5 

17508 

98455 

19224 

98135 

20933 

97784 

22637 

97404 

24333 

96994 
96987 

55 

6 

17537 

98430 

19232 

98129 

20962 

97778 

22665 

97398 

24362 

54 

7 

17565 

98445 

19281 

98124 

20990 

97772 

22693 

97391 

24390 

96980 

53 

8 

17394 

98440 

19309 
1933^ 

98118 

21019 

97766 

22722 

97384 

24418 

96973 

52 

9 

17623 

98435 

98112 

21047 

97760 

22750 

97378 

24446 

96966 

5i 

lo 

17651 

98430 

19366 

98107 

21076 

97754 

22778 

97371 

24474 

96959 

5o 

I! 

17680 

98425 

19395 

98101 

21104 

97748 

22807 

97365 

245o3 

96952 

49 

\7 

17708 

98420 

19423 

98096 

2Il32 

97742 

22835 

97358 

24531 

96945 

48 

i3 

17737 

98414 

19452 

98090 

21161 

97733 

22863 

97351 

24559 

96937 

47 

14 

17766 

98409 

19481 

98084 

21180 

97729 

22892 

97345 

24587 

96930 

46 

13 

17794 

98404 

19509 

98079 

21218 

97723 

22920 

97338 

24615 

96923 

45 

i6 

17823 

98399 
98394 

19538 

98073 

21246 

97717 

22948 

97331 

24644 

96916 

44 

17 

17852 

19366 

98067 

21275 

97711 

22977 

97325 

24672 

96909 

43 

i8 

17880 

98389 

19395 

98061 

2i3o3 

97705 

23oo5 

97318 

24700 

96902 

42 

'9 

17909 

98383 

19623 

98036 

2i33i 

97698 

23o33 

97311 

24728 

96894 

41 

20 

17937 

98378 

19652 

98050 

2i36o 

97692 

23062 

97304 

24756 

96887 

40 

21 

17966 

98373 

19680 

98044 

21 388 

97686 

23090 

97298 

24784 

96880 

39 

22 

17993 

98368 

19709 

98039 

21417 

97680 

23ii8 

97291 

248i3 

96873 

38 

23 

18023 

98362 

19737 

98033 

21445 

97673 

23i46 

97284 

24841 

96866 

37 

24 

l8032 

98357 

19766 

98027 

21474 

97667 

23175 

97278 

24869 

96858 

36 

23 

18081 

98352 

19794 

98021 

2l502 

97661 

232o3 

97271 

24897 

96851 

35 

26 

18109 

9^?^'' 

19823 

98016 

2i53o 

97633 

2323l 

97264 

24925 

96844 

34 

27 

iSi38 

98341 

19851 

98010 

21559 

97648 

23260 

97237 

24954 

96837 

33 

28 

18166 

98336 

19880 

98004 

21587 

97642 

23288 

97231 

24982 

96S29 

33 

?9 

18193 

98331 

19908 

97998 

21616 

97636 

233 16 

97244 

23010 

96822 

Si 

3o 

18224 

98325 

19937 

97992 

21644 

97630 

23345 

97237 

25o38 

96S15 

So 

3i 

18252 

98320 

19965 

97987 

21672 

97623 

23373 

9723o 

25o66 

96807 

29 

3i 

18281 

983 13 

19994 

97981 

21701 

97617 

23401 

97223 

25094 

96800 

28 

33 

18309 

98310 

20022 

97975 

21729 

97611 

23429 

97217 

25l22 

96793 

27 

?i 

i8338 

98304 

2oo5i 

97969 

21758 

97604 

23458 

97210 

25i5i 

96786 

26 

35 

1 8367 

98299 

20079 

97963 

21786 

97598 

23486 

97203 

23179 

96778 

23 

36 

18395 

98294 

20108 

97958 

21814 

97592 

235i4 

97196 

25207 

96771 

24 

?Z 

18424 

98288 

2oi36 

97952 

21843 

97385 

23542 

97189 

23235 

96764 

23 

38 

18452 

98283 

2oi65 

97946 

21871 

97579 

23571 

97182 

25263 

96756 

23 

39 

18481 

98277 

20193 

97940 

21899 

97573 

23599 

97176 

25291 

96749 

31 

40 

i85o9 

^8272 

20222 

97934 

21928 

97566 

23627 

97169 

25320 

96742 

30 

41 

i8538 

98267 

2025o 

97928 

21936 

97560 

23656 

97162 

25348 

96734 

10 

42 

18567 

98261 

20279 

97922 

21985 

97553 

23684 

97155 

25376 

96727 

j8 

43 

18595 

98255 

2o3o7 

97916 

220l3 

97547 

23712 

97148 

23404 

96719 

17 

44 

18624 

98230 

2o336 

97910 

22041 

97341 

23740 

97141 

25432 

96712 

16 

i5 

45 

18652 

98245 

2o364 

97905 

22070 

97534 

23769 

97134 

25460 

96705 

46 

18681 

98240 

20393 

97899 

22098 

97528 

23797 

97127 

25488 

96697 

14 

47 

18710 

98234 

20421 

97893 

22126 

97521 

23825 

97120 

255i6 

96690  1  i3  j 

48 

18738 

98229 

2045o 

97887 

22l55 

975i5 

23853 

97113 

25545 

96682 

12 

49 

18767 

9^'^^ 

20478 

97881 

22l83 

97508 

23882 

97106 

25573 

96675 

II 

DO 

18795 

98218 

2o5o7 

97873 

22212 

97502 

23910 

97100 

2  3601 

96667 

10 

5i 

18824 

98212 

2o535 

97S69 

22240 

97496 

23938 

97093 

25629 

96660 

I 

52 

18852 

98207 

20563 

97863 

22268 

97480 
97483 

23966 

97086 

25657 

96653 

53 

18881 

98201 

20592 

97837 

22297 

23995 

97079 

25685 

96645 

7 

54 

18910 

98196 

20620 

97851 

22325 

97476 

24023 

97072 

25713 

96638 

6 

55 

18938 

98190 

20649 

97845 

22353 

97470 

24o5i 

97065 

25741 

96630 

5 

56 

18967 

98185 

20677 

97839 

22382 

97463 

24070 

97o58 

25769 

96623 

4 

^1 

18995 

98179 

20706 

97833 

22410 

97457 

24108 

97o5i 

25798 

96615 

3 

58 

19024 

98174 

20734 

97827 

22438 

97450 

24i36 

97044 

25826 

96608 

s 

59 

19052 

98168 

20763 

97821 

22467 

97444 

24164  97037 

25854 

96600 

I 

60 

19081 

98163 

20791 

97815 

22495 

97437 

24192  97o3o 

25882 

96393 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

/ 

/ 

79° 

'TS^ 

77° 

^e" 

"IR" 

Table  III.      NATURAL  SINES  AND  COSINES.           67  | 

/ 

15° 

16° 

17° 

18° 

19° 

/ 

60 

Sine.  1  Cosine. 

Sine. 

Cosine. 

Siiie. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

0 

25882 

96593 

27564 

96126 

29287 

9563o 

80902 

95106 

32557 

94552 

I 

25910 

96585 

27592 

96118 

29265  ;  95622 

80929 

95097 

32584 

94542 

U 

2 

25938 

96578 

27620 

96 1 1 0 

29298 

93618 

80957 

95088 

82612 

94583 

3 

25966 

96570 

27648 

96102 

29J21 

956o5 

80985 

95079 

3:689 

94528 

57 

4 

23994 

96562 

27676 

96094 
96086 

29848 

95596 

3ioi2 

95070 

82667 

945 14 

56 

5 

26022 

96555 

27704 

29876 

95588 

81040 

95061 

82694 

945o4 

55 

6 

26o5o 

96547 

27731 

96078 

29404 

95579 

81068 

95o52 

82722 

94495 
94483 

54 

7 

26079 

96540 

27739 

96070 

29482 

95571 

81095 

95043 

32749 

53 

8 

26107 

96532 

27787 

96062 

29460 

g5562 

31128 

95o33 

32777 
82804 

94476 

5a 

9 

26135 

96524 

27815 

96054 

29487 

95554 

3ii5i 

95024 

94466 

5i 

10 

26163 

96517 

27843 

96046 

29515 

95545 

81178 

950 1 5 

82882 

94457 

5o 

II 

26191 

96509 

27871 

96087 

29543 

95536 

81206 

95006 

82859 

94447 
9448S 

g 

12 

26219 

96502 

27899 

96029 

29571 

95528 

81288 

9499"? 

82887 

i3 

26247 

96404 

27927 

96021 

29599 

95519 

81261 

94988 

32914 

94428 

47 

i4 

26275 

96486 

27955 

96018 

29626 

95511 

81289 

94979 

82942 

94418 

46 

i5 

263o3 

96479 

27983 

96005 

29634 

95502 

3i3i6 

94970 

82969 

94409 

45 

i6 

26331 

96471 

2801 1 

95997 

29682 

95498 

81844 

94961 

82997 

94899 

44 

'7 

26359 

96463 

28039 

95989 

29710 

95485 

81872 

94952 

88024 

94890 

43 

i8 

26387 

96456 

28067 

95981 

;  29787 

95476 

31899 

94943 

88o5i 

94880 

42 

'9 

26415 

96448 

28095 

95972 

29765 

95467 

3i427 

94933 

88079 

94870 

41 

20 

26443 

96440 

28123 

95964 

29798 

95459 

81454 

94924 

38 106 

94861 

40 

21 

26471 

96433 

28i5o 

95956 

29821 

95450 

81482 

94915 

88184 

948  5 1 

39 

22 

265oo 

96425 

28178 

95948 

29849 

95441 

8i5io 

94006 
94897 

38 161  94842 

38 

23 

26528 

96417 

28206 

95940 

29876 

95488 

81537 

88189  94882 

37 

24 

26556 

96410 

28234]  93931 

29904 

95424 

3i565 

94888 

88216  94822 

36 

23 

26584 

96402 

28262 

93928 

29982 

95415 

81593 

94878 

83244 

94818 

85 

26 

26612 

96394 

28290 

93913 

29960 

93407 

81620 

94869 

33271 

94808 

34 

27 

26640 

96386 

283i8 

95907 

29987 

95808 
95889 

81648 

94860 

88298 

94208 
94284 

33 

28 

26668 

96379 

28346 

95^:98 

3ooi5 

81675 

94851 

88826 

32 

29 

26696 

96371 

28374 

95890 

80048 

95880 

81708 

94842 

33858 

94274 

81 

3o 

26724 

96363 

28402 

95882 

80071 

95372 

31780 

94882 

88881 

94264 

3o 

3i 

26752 

96355 

28429 

95874 

80098 

95863 

81758 

94823 

33408 

94254 

29 

32 

26780 

96347 

28457 

95865 

80126 

95354 

81786 

94814 

83486 

94245 

28 

3  J 

26808 

96340 

28485 

95857 

3oi54 

95345 

3i8i8 

94805 

88468 

94235 

27 

34 

26836 

96332 

285 1 3 

95849 

80182 

95887 

3i84i 

94795 

88490 

94225 

26 

35 

26S64 

96324 

28541 

g584i 

80209 

95828 

3iS68 

94786 

885i8 

94215 

25 

36 

26892 

96316 

28569 

95832 

80287 

95819 

81896 

9-i777 

88545 

94206 

24 

J7 

26920 

96308 

28597 

95824 

80265 

95310 

81928 

94768 

33578 

94196 

28 

38 

26948 

9O301 

28625 

95816 

80292 

95801 

81951 

94758 

836oo 

94186 

22 

39 

26976 

96203 

28652 

95807 

80820  95298 

81979 

94749 

88627 

94176 

21 

40 

27004 

96285 

28680 

93799 

80848  95284 

82006 

94740 

83655 

94167 

20 

4i 

27032 

96277 

2S708 

95791 

80876 

95275 

82084 

94780 

83682 

94137 

19 

42 

27060 

96269 

28736 

93782 

80408 

95266 

32061 

9*4721 

33710:94147 

18 

43 

27088 

96261 

28764 

93774 

80481 

95257 

82089 

94712 

88787  94187 

17 

44 

27116 

96253 

28792 

95766 

80439  95248 

82116 

94702 

88764  94127 

16 

45 

27144 

96246 

2S820 

95757 

80486 

95240 

82144 

94698 

88792 

94110 

i5 

46 

27172 

96238 

28847 

95749 

3o5i4 

93281 

82171 

94684 

88819 

94108 

14 

47 

27200 

96230 

28875 

95740 

3o542 

95222 

82199 

94674 

33846 

94098 

i3 

48 

27228 

96222 

28908 

93782 

80570 

93218 

82227 

94665 

88874 

94088 

12 

49 

27256 

96214 

28981 

95724 

80597 

95204 

32254 

94656 

88901 

94078 

II 

5o 

27284 

96206 

28959 

95715 

80625 

95195 

82282 

94646 

88929 

94068 

10 

5i 

27312 

96198 

28987 

95707 

3o658  95186 

82809 

94687 

88956 

94o58 

t 

52 

27340 

96190 

2i>Ol5 

93698 

3o68o  9517-7 
80708  95168 

82887 

94627 

88988 

94049 

53 

27368 

961 82 

29042 

95690 

82864 

94618 

34011 

94089 

7 

54 

27396 

96174 

29070 

95681 

30786  95159 

82892 

94609 

84088 

94029 

6 

55 

27424 

96 1 66 

29098 

93673 

80768  95i5o 

82419 

94599 

84065 

94019 

5 

56 

27452 

96 1 58 

29126 

95064 

80791  95r42 

82447 

94300 
94580 

84098 

94009 

4 

37 

27480 

96150 

29154 

95656  j  80819  1  95188 

82474 

84120 

98909 
98989 

3 

58 

27508 

96142 

29182 

93647 

80846  95124 

82502 

94371 

84147 

2 

09 

27536 

96134 

29209 

95689 

80874 

95ii5 

82529 

94561 

84175 

98979 

I 

60 

27564 

96126 

29287 

95680  1  80902 

95106 

82557 

94532 

84202 

98969 

0 

Cosine.   Sine. 

Cosine. 

Sine. 

Cosine.   Sine. 

Cosine. 

Sine. 

Cosine.  Sine. 

/ 

/ 

743 

73° 

'72° 

71° 

70° 

68            NATURxVL  SINES  AXD  COSINES.      Table  III.  | 

/ 

20° 

010     1 
_1      1 

22° 

23° 

24° 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosiiie. 

0 

34202 

93969 

35837 

93358 

37461 

92718 

39073 

92o5o 

40674 

91355 

60 

1 

34229 

93959 

35864 

93348 

37488 

92707 

39100 

92o3o 

40700 

91343 

59 

2 

34257 

93949 

35891 

93337 

37315 

92697 

39127 

92028 

40727 

9i33i 

58 

3 

34284 

93939 

35918 

93327 

37542 

92686 

39153 

92016 

40753 

91319 

57 

4 

343  11 

93929 

35945 

93316 

37569 

92675 

39180 

92005 

40780 

91307 

56 

5 

34339 

93919 

35973 

93306 

37595 

92664 

39207 

91904 
91982 

40806 

91205 

55 

6 

34366 

93909 

36000 

93293 

37622 

92653 

39234 

4o833 

91283 

54 

7 

34393 

93899 

36027 

93285 

07649 

92642 

39260 

91971 

40860 

91272 

53 

8 

34421 

93889 

36o54 

93274 

37676 

92631 

39287 

91959 

40886 

91260 

52 

9 

34448 

93879 

36o8i 

93264 

37703 

92620 

39314 

91948 

40913 

91248 

5i 

10 

34475 

93869 

36io8 

93253 

37730 

92609 

39341 

91936 

40939 

91236 

5o 

11 

345o3 

93809 

36i35 

93243 

37757 

92598 

39367 

91925 

40966 

91224 

49 

12 

3453o 

93849 

36162 

93232 

37784 

92587 

39394 

91914 

40992 

91212 

48 

i3 

34557 

93839 

36190 

93222 

37811 

92576 

39421 

91002 
91891 

41019 

91200 

47 

U 

34584 

93829 

36217 

93211 

37838 

92565 

39448 

41045 

91188 

46 

ID 

34612 

93819 

36244 

93201 

37865 

92554 

39474 

91879 

41072 

91176 

45 

i6 

34639 

93809 

36271 

93190 

37892 

92543 

39501 

91868 

41098 

91164 

44 

17 

34666 

93799 

36298 

93180 

37919 

92532 

39528 

91856 

41125 

91152 

43 

i8 

34694 

93789 

36325 

93169 

37946 

92521 

39555 

91845 

4ii5i 

91140 

42 

'9 

34721 

93779 

36352 

93159 

37973 

92510 

39581 

91833 

41178 

91 1 28 

41 

20 

34748 

93769 

36379 

93148 

37999 

92400 

39608 

91822 

41204 

91116 

40 

21 

34775 

93759 

36406 

93137 

38026 

92488 

39635 

91810 

4i23i 

91 1 04 

39 

22 

34803 

93748 

36434 

93127 

38o53 

92477 

39661 

91799 

41257 

91092 

38 

23 

3483o 

93738 

36461 

93116 

38o8o 

92466 

39688 

91787 

41284 

91080 

37 

24 

34857 

93728 

36488 

93106 

38107 

92455 

39715 

91775 

4i3io 

91068 

36 

25 

34884 

93718 

365 1 5 

93095 

38i34 

92444 

39741 

91764 

41337 

9io56 

35 

26 

34912 

93708 

36542 

93084 

38i6i 

92432 

39768 

91752 

41363 

91044 

34 

27 

34939 

93698 
93688 

36569 

93074 

38i88 

92421 

39795 

91741 

41390 

91032 

33 

28 

34966 

36596 

93o63 

38215 

92410 

39822 

91729 

41416 

91020 

32 

2Q 

34993 

93677 

36623 

93o52 

38241 

92399 

39848 

91718 

41443 

91008 

3i 

3o 

35o2i 

93667 

3665o 

93c42 

38268 

92388 

39875 

91706 

41469 

90996 

3o 

3i 

35048 

93657 

36677 

93o3i 

38295 

92377 

39902 

91694 

41496 

90984 

29 

32 

35075 

93647 

36704 

93020 

38322 

92366 

39928 

91683 

4l522 

90972 

28 

33 

35io2 

93637 

36731 

93010 

38349 

92355 

39955 

91671 

41549 

90960 

27 

34 

35i3o 

93626 

36758 

92999 

38376 

92343 

39982 

91660 

4157? 

90948 

26 

35 

35i57 

93616 

36785 

92988 

384o3 

92332 

40008 

91648 

41602 

90936 

25 

36 

35i84 

93606 

36812 

92978 

38430 

92321 

4oo35 

91636 

41628 

90924 

24 

37 

35211 

93596 

36839 

92967 

38456 

923io 

40062 

91625 

4i655 

9091 1 

23 

38 

35239  93585  1 

36867 

92956 

38483 

92209 

40088 

91613 

41681 

90899 

22 

39 

35266 

93375 

36894 

92945 

385io 

92287 

4oii5 

91601 

41707 

90887 

21 

40 

35293 

93565 

36921 

92935 

38537 

92276 

40141 

91590 

41734 

90875 

20 

41 

35320 

93555 

36948 

92924 

38564 

92265 

40168 

91578 

41760 

90863 

19 

42 

35347 

93544 

36975 

92913 

38591 

92254 

40195 

91566 

41787 

9o85i 

18 

43 

35375 

93534 

37002 

92902 

38617 

92243 

40221 

91555 

4i8i3 

90839 

\l 

44 

35402 

93524 

37029 

92892 

38644 

92231 

40248 

91543 

41840 

90826 

45 

35429 

93514 

37056 

92881 

38671 

92220 

40275 

9i53i 

41866 

90814 

ID 

46 

35456 

935o3 

37083 

92870 

38698 

92209 

4o3oi 

91519 

41892 

90802 

14 

47 

35484 

93493 
93483 

37110 

92859 

38725 

92ip§ 

4o328 

9i5o8 

41919 
41945 

90790 

i3 

48 

35511 

37137 

92849 

38752 

92186 

40355 

91496 

90778 

12 

49 

35538 

93472 

37164 

92838 

38778 

92175 

4o38i 

91484 

41972 

90766 

11 

5o 

35565 

93462 

37191 

92827 

388o5 

92164 

40408 

91472 

41998 

90753 

10 

5i 

355g2 

93452 

37218 

92816 

38832 

92l52 

40434 

91461 

42024 

90741 

9 

52 

35619 

93441 

37245 

92805 

38859 

92141 

40461 

91449 

42o5i 

90729 

8 

53 

35647 

93431 

37272 

92704 

38886 

92i3o 

40488 

91437 

42077 

90717 

I 

54 

35674 

93420 

37299 

927S4 

38912 

92119 

4o5i4 

91425 

42104 

90704 

55 

35701 

93410 

37326 

92773 

38939 

92107 

40541 

91414 

42i3o 

90692 

5 

56 

35728 

93400 

37353 

92762 

38966 

92006 
92085 

40567 

91402 

42156 

90680 

4 

u 

35755 

93389 

37380 

92751 

38993 

40594 

91390 

42183 

90668 

3 

35782 

93379 

37407 

92740 

39020 

92073 

40621 

91378 

42209 

90655 

3 

5g 

358io 

93368 

37434 

92720 

39046 

92062 

40647 

91366 

42233 

90643 

I 

6o 

35837 

93358 

37461 

92718 

39073 

9io5o 

40674 

91355 

42262 

9063 1 

0 

/ 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

/ 

69° 

68° 

67° 

66° 

65° 

TAbi-E  III.      NATURAL  SINES  AND  COSINES.           69  | 

/ 

25° 

26° 

27° 

28° 

29° 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

1 
Cosine. 

Sine. 

Cosine. 

0 

42262 

9063 1 

43837 
43863 

89879 

45399 

89101 

46947 

88295 

48481 

87462 

60 

1 

42288 

90618 

89867 

45423 

89087 

46973 

88281 

485o6 

87448 

59 

2 

423 1 5 

90606 

43889 

89834 

45451 

89074 

46999 

88267 

48532 

87434 

58 

3 

42341 

90594 

43916 

89841 

45477 

89061 

47024 

88234 

48337 

87420 

57 

4 

42367 

9o582 

43942 

89828 

455o3 

89048 

47o5o 

88240 

48583 

87406 

56 

5 

42394 

90569 

43968 

89816 

43329 

89035 

47076 

88226 

48608  87391 

55 

6 

42420 

90557 

43994 

89803 

45554 

89021 

47101 

88213 

48634  87377 

54 

I 

42446 

90545 

44020 

89790 

45580 

89008 

47127 

88199 

48659 

87363 

53 

42473 

90532 

44046 

89777 

45606 

88995 

47153 

88i83 

48684 

87349 

52 

9 

42499 

90520 

44072 

89764 

45632 

88981 

47178 

88172 

48710 

87335 

5i 

iO 

42323 

9o5o7 

44098 

89752 

45653 

88968 

47204 

88 1 58 

48735 

87321 

5o 

II 

42552 

90493 

44124 

80739 

45684 

88955 

47229 
47255 

88144 

48761 

87306 

49 

12 

42578 

90483 

44i5i 

89726 

45710 

88942 

88i3o 

48786 

87292 

43 

i3 

42604 

90470 

44177 

89713 

45736 

88928 

47281 

88117 

48811 

87278 

47 

i4 

42631 

90438 

442o3 

89700 

43762 

88915 

47306 

88io3 

48837 

87264 

46 

i5 

42657 

90446 

44229 

89687 

43787 

88902 

47332 

88089 

4_8862 

87250 

43 

i6 

42683 

90433 

44255 

89674 

458i3 

88888 

47358 

88075 

488S8 

87235 

44 

17 

42709 

90421 

44281 

89662 

45839 

88875 

47383 

88062 

48913 

87221 

43 

18 

42736 

90408 

44307 

89649 

45863 

88862 

47409 

88048 

48933 

87207 

42 

'9 

42762 

Qo3g6 

44333 

89636 

45891 

88848 

47434 

88o34 

48964 

87.93 

41 

20 

42788 

90383 

44359 

89623 

45917 

88835 

47460 

88020 

48989 

87178 

40 

21 

428 1 5 

90371 

4438D 

89610 

45942 

88822 

47486 

88006 

49014 

87164 

3q 

22 

42841 

90358 

444 1 1 

89597 

45968 

88808 

47511 

87993 

49040 

87150 

33 

23 

42867 

90346 

44437 

89584 

45994 

88795 

47537 

87979 

49065 

87136 

37 

24 

42894 

90334 

44464 

89571 

46020 

88782 

47562 

87963 

49090 

87121 

36 

25 

42920 

9o32i 

44490 

89558 

46046 

88768 

47588 

87951 

49116 

87107 

35 

26 

42946 

90309 

445 1 6 

89545 

46072 

88755 

47614 

87937 

49141 

87093 

34 

11 

42972 

90296 

44542 

89532 

46097 

88741 

47639 

87923 

49166 

87079 

33 

42999 

90284 

44568 

89519 

46123 

88728 

47663 

87909 

49192 

87064 

32 

29 

43o2  3 

90271 

44594 

89506 

46149 

88715 

47690 

87896 

49217 

87050 

3i 

3o 

43o5i 

90259 

44620 

89493 

46173 

88701 

47716 

87882 

49242 

87036 

3o 

3i 

43077 

90246 

44646 

89480 

46201 

88688 

47741 

87868 

49268 

87021 

29 

32 

43104 

90233 

44672 

89467 

46226 

88674 

47767 

87854 

49293 

87007 

23 

33 

43i3o 

90221 

44698 

89454 

46252 

88661 

47793 

87840 

49318 

86993 

27 

34 

43156 

90208 

44724 

89441 

46278 

88647 

47818 

87826 

49344 

86978 

20 

35 

43182 

90196 

44730 

89428 

463o4 

88634 

47844 

87812 

49369 

86964 

25 

36 

43209 

90183 

44776 

89413 

46330 

88620 

47869 

87798 

49394 

86949 

24 

37 

43233 

90171 

44802 

89402 

46355 

88607 

47895 

87784 

49419 

86933 

23 

33 

43261 

90i58 

44828 

89389 

4638i 

88593 

47920 

87770 

49443 

86921 

22 

39 

43287 

90146 

44854 

89376 

46407 

88580 

47946 

87756 

49470 

86906 

21 

4o 

433 1 3 

90133 

44880 

89363 

46433 

88566 

47971 

87743 

49495 

86892 

20 

41 

43340 

90120 

44906 

89350 

46458 

88553 

47997 

87729 

49521 

86878 

19 

42 

43366 

90108 

44932 

89337 

46484 

83539 

48022 

877,5 

49546 

86863 

i3 

43 

43392 

90095 

44958 

89324 

465 10 

88526 

48048 

87701 

49571 

86849 

17 

44 

43418 

90082 

44984 

89311 

46536 

88512 

48073 

87687 

49396 

86834 

16 

45 

43445 

90070 

45oio 

89298 

46361 

88499 

48099 

87673 

49622 

86820 

i5 

46 

43471 

90057 

45o36 

89285 

46587 

88485 

48124 

87659 

49647 

868o5 

14 

47 

43497 

90043 

45062 

89272 

46613 

88472 

481 30 

87645 

4967  a 

86791 

i3 

48 

43323 

90o32 

45o88 

89259 

46639 

88458 

48175 

87631 

49697 

86777 

12 

49 

43549 

90019 

45ii4 

89243 

46664 

88445 

48201 

87617 

49723 

86762 

II 

30 

43575 

90007 

43 140 

89232 

46690 

88431 

48226 

87603 

49748 

86748 

JO 

5i 

43602 

89994 

43166 

89219 

46716 

88417 

48252 

87589 

49773 

86733 

I 

52 

43628 

89981 

45192 

89206 

46742 

88404 

48277 

87575 

49798 

86719 

53 

43654 

89968 

45218 

89103 
89180 

46767 

88390 

483o3 

87561 

49824 

86704 

7 

•54 

43680 

89956 

43243 

46793 

88377 

48328 

87546 

49849 

86690 

6 

55 

43706 

89943 

45269 

89167 

46819 

88363 

48354 

87532 

49874 

86675 

5 

56 

43733 

89930 

43293 

89153 

46844 

88349 

48379 

873x8 

49899 

86661 

4 

ll 

43759 

89918 

45321 

89140 

46870 

88j36 

48400 

87504 

49924 

86646 

3 

43783 

89905 

45347 

89127 

46896 

88322 

48430 

87490 

49950 

8663a 

3 

59 

43811 

89892 

45373 

89114 

46921 

883o8 

48456 

87476 

49975 

86617 

I 

60 

43837 

89879 

45399 

89101 

46947 

88295 

48481 

87462 

5oooo 

866o3 

0 

/ 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

> 

64° 

63° 

62° 

61° 

60° 

70            NATURAL  SINES  AND  COSINES.      Table  III. 

0 

30° 

31° 

32° 

33° 

34° 

/ 

Sine.  Cosine. 

Sine.  Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine.  Cosine. 

5oooo  866o3 

5l5o4  85717 

02992 

848o5 

54464 

83867 

55919  82904  I  60  1 

1 

5oo25 

86588 

5i529  85702 

53017 

84789 

54488 

8385 1 

55943 

82887  5g  1 

2 

5oo5o 

86573 

5io54  85687 

53o4i 

B4774 

545 1 3 

83835 

55968 

82871 

58 

3 

50076 

86509 

5i579  80672 

53o66 

84759 

54537 

83819 

55992 

82800 

57 

4 

Soioi 

86544 

5 1604 

85657 

53091 

84743 

5456 1 

83804 

56oi6 

8283g 

56 

5 

00126 

86530 

51628 

85642 

53!i5 

84728 

54586 

83788 

56o4oi  828:22 

55 

6 

5oi5i 

865i5 

5i653 

85627 

53 140 

84712 

54610 

83772 

56064  82806 

54 

7 

50176 

86501 

51678 

85612 

53 1 64 

84697 

54635 

83706 

56o88  1  82790  53  1 

8 

5o20I 

86486 

5i7o3 

85597 

53189 

84681 

54659 
54683 

83740 

56ii2  82773 

52 

9 

50227 

86471 

51728 

85582 

53214 

84666 

83724 

56i36  '  82757 

01 

10 

50202 

86457 

51753 

85567 

53238 

8465o 

54708 

83708 

56i6o  82741 

5o 

II 

50277 

86442 

5.778 

8555 1 

53263 

84635 

54732 

836q2 

56iS4  82724 

49 

13 

5o3o2 

86427 

5i8o3 

85536 

53288 

84619 

54756  83676 

56203  82708 

48 

i3 

5o327 

86413 

51828 

85521 

53312 

84604 

54781 

8366o 

56232  ]  82692  1  47 

U 

5o352 

86398 

5i852 

80006 

53337 

84588 

548o5 

83645 

56206  82670  46 

i5 

5o377 

86384 

51877 

85491 

53361 

84573 

54829 

83629 

56280  1  82659  '  40 

i6 

5o4o3 

86369 

51902 

85476 

53386 

84557 

54804 

836i3 

563o5  82643  i  44 

17 

50428 

86354 

01927 

85461 

53411 

84542 

54878 

83597 

56329  1  82626 

43 

18 

5o453 

86340 

51952 

80446 

53435 

84526 

04902 

83581 

56353  :  82610 

42 

19 

50478 

86325 

01977 

8043 1 

53460 

84511 

54927 

83565 

56377  1  82593 

41 

20 

5o5o3 

86310 

52002 

854i6 

53484 

84495 

54951 

83549 

06401 

82577 

40 

21 

5o528 

86295 

52026 

85401 

53509 

84480 

54975 

83533 

56425 

82061 

39 

22 

5o553 

86281 

O2o5l 

85385 

53534 

84464 

54999 

83517 

56449 

82544 

38 

23 

50578 

86266 

52076 

80370 

53558 

84448 

55o24 

835oi 

56473 

82528 

37 

24 

5o6o3 

86201 

52101 

85355 

53583 

84433 

55048 

83485 

56497 

82011 

3t) 

25 

50628 

86237 

52126 

85340 

53607 

84417 

55072 

83469 

56521 

82490 

35 

26 

5o654 

86222 

52i5i 

85325 

53632 

84402 

55o97 

83453 

56545 

82478!  34 

27 

50679 

86207 

52175 

853 10 

53656 

84386 

50121 

83437 

56569 

82462  33 

23 

50704 

86192 

52200 

852g4 

53681 

84370 

55i45 

83421 

56593 

82446  32 

29 

00729 

86178 

02225 

80279 

53705 

84355 

55169 

834o5 

56617 

82429  '  3i 

3o 

50704 

861 63 

5225o 

80264 

53730 

84339 

55194 

83389 

56641 

824.3  1  3o 

3 1 

50779 

86148 

02275 

85249 

53754 

84324 

55218 

83373 

56665 

82396 

29 

32 

5o8o4 

86i33 

52299 

85234 

53779 

843o8 

55242 

83356 

56689 !  82380 

28 

33 

00829 

86119 

52324 

85218 

53804 

84292 

55266 

83340 

56713 

82363 

27 

34 

5oS54 

86104 

02349 

85203 

53828 

84277 

55291 

83324 

56736 

82347 

26 

35 

00879 

86089 

52374 

85x88 

53853 

84261 

553 1 5 

833o8 

56t6o 

82330 

25 

36 

00904 

86074 

52399 

85 173 

53877 

84245 

55339 

83292 

56784 

82314 

24 

37 

50929 

86059 

02423 

85i57 

53902 

84230 

55363 

83276 

06808 

82297 

23 

38 

50954 

86040 

02448 

85i42 

53926 

84214 

55388 

83260 

56832 

82281 

22 

39 

50979 

86o3o 

52473 

85i27 

53901 

84198 

55412 

83244 

56856 

82264 

2. 

40 

01004 

8601 5 

52498 

85ii2 

53975 

84182 

55436 

83228 

56880 

82248 

20 

41 

Oi02g 

86000 

52522 

85096 

54000 

84167 

55460 

83212 

56904 

82231 

19 

42 

5io54 

85985 

52547 

85o8i 

54024 

84101 

55484 

83195 

56928 

82214 

18 

43 

51079 

80970 

52572 

85o66 

04049 
5407.3 

84135 

55509 

83 1 79 

56952 

82198 
82181 

17 

44 

5iio4 

85956 

52597 

85o5i 

84120 

55533 

83 1 63 

06976 

16 

45 

51129 

85941 

52621 

8oo35 

54097 

84104 

55557 

83 147 

57000 

82165 

i5 

46 

5ii54 

85926 

52646 

85020 

54122 

84088 

5558i 

83i3i 

5/024 

82148 

14 

47 

01179 

80911 

52671 

8ooo5 

54146 

84072 

556o5 

83ii5 

57047 

82132 

i3 

48 

5i2o4 

80896 

52696 

84989 

54171 

84057 

5563o 

83098 
83082 

57071 

821.5 

12 

49 

5i22g 

80881 

52720 

84974 

54195 

84041 

55654 

57095 

82098 

II 

DO 

5 1 254 

80866 

52740 

84959 
84943 

54220 

84025 

50678 

83o66 

07119 

82082 

10 

5i 

51279 

8o85i 

52770 

54244 

84009 

50702 

83o5o 

57143 

82065 

^ 

02 

5i3o4 

85836 

52794 

84928 

54269 
5429J 

83994 

55726 

83o34 

57.67 

82048 

53 

5i329 

80821 

52819 

849  >  3 

83978 

55750 

83017 

57191 

82032 

7 

54 

5i354 

80806 

52844 

84897 

543 1 7 

83962 

55775 

83ooi 

57210  82010 

6 

55 

5i379 

85792 

52869 

84882 

54342 

83946 

50799 

829S0 

57238  81999 

5 

■S6 

01404 

85777 

52893 

84866 

54366 

83930 

55823  82969 

57262  81982 

4 

u 

01429 

-85762 

52918 

8485 1 

04391 

83910 

55847  82953 

57286  81965 

3 

51454 

85747 

52943 

84836 

04410 

83899 

55871  82936 

57310  81949 

2 

59 

5 1479 

85732 

52967 

84820 

04440 

83883 

55895  82920 

57334  S1932 

I 

60 

5ioq4  1  85717 

52992 

84805 

I  54464 

83867 

55919  82904 

57358  81915 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine.   Sine. 

Cosine.   Sine. 

Cosine. 

Sine. 

/ 

/ 

69= 

58° 

5,0 

56° 

65° 

Table  III. 

NATURAL 

SINES  ANll 

COSINES. 

71 

35° 

36° 

31°   ! 

38° 

39° 

/ 
60 

Sine.  Cosine. 

Sine.  Cosine. 

1 

Sine.  Cosine. 

Sine. 

Cosine. 

Sine.  Cosine. 

0 

57358 

81915 

58779  1  80902 

60182 

79864 

61 566 

78801 

62932 

77715 

I 

57381 

81890 

81882 

58802  8oS85 

6o2o5 

79846 

61589 

78783 

62955 

77696 

59 

3 

57405 

58826 

80867 

60228 

79829 

61612 

78765 

62977 

77678 

58 

3 

57429 
57453 

81 865 

58849 

8o85o 

602  5 1 

79811 

6i635 

78747 

63ooo 

77660 

57 

4 

81848 

58873 

80833 

60274 

79793 

61 658 

78729 

63022 

77641 

56 

5 

57477 

8i832 

58896 

80816 

60298 

79776 

61681 

787 1 1 

63o45 

77623 

55 

6 

57501 

8i8i5 

5S920 

80799 
80782 

6o32i 

79738 
79741 

6. -704 

78694 

63o68 

77603 

54 

I 

57524 

81798 

58943 

6o344 

61726 

78676 

63090 

77586 

53 

57548 

81782 

58967 

80765 

60367 

79723 

61749 

78658 

63ii3 

77568 

52 

9 

57572 

81765 

58990 

80748 

60390 

79706 

61772 

78640 

63i35 

77530 

5 1 

10 

57596 

81748 

59014 

80730 

60414 

79688 

61795 

78622 

63i58 

77331 

5o 

11 

57619 

81731 

59037 

80713 

60437 

79671 

61818 

78604 

63 1 80 

775i3 

49 

12 

57643 

81714 

09061 

80696 

60460 

79653 

61841 

78586 

632o3 

77494 

48 

i3 

57667 

81698 

59084 

80679 

60483 

79635 

61864 

78568 

63225 

77476 

47 

14 

57691 

81681 

59108 

80662 

6o5o6 

79618 

61887 

78550 

63248 

77438 

46 

ID 

57715 

81664 

59131 

80644 

60529 

79600 

61909 

78332 

63271 

77439 

43 

i6 

57738 

81647 

59154 

80627 

6o553 

79583 

61932 

78314 

63293 

77421 

44 

\l 

57762 

8i63i 

59178 

80610 

60576 

79565 

61955 

78496 

633 16 

77402 

43 

57786 

81614 

59201 

80393 

60099 

79347 

61978 

78478 

63338 

77384 

42 

•9 

57810 

81597 

59225 

80376 

60622 

79530 

62001 

78460 

63361 

77366 

41 

20 

57833 

8i58o 

5924S 

80558 

60645 

79512 

62024 

78442 

63383 

77347 

40 

21 

57837 

81 563 

59272 

8o54i 

60668 

79494 

62046 

78424 

63406 

77329 

39 

22 

57881 

81546 

59295 

8o524 

60691 

79477 

62069 

78405 

63428 

77310 

38 

23 

57904 

8i53o 

59318 

8o5o7 

60714 

79439 

62092 

783S7 

63431 

77292 

37 

24 

57928 

8i5i3 

59342 

80489 

60738 

79441 

621 15 

78369 

63473 

77273 

30 

23 

57952 

81496 

59365 

80472 

60761 

79424 

62138 

78351 

63496 

77233 

35 

26 

'J7976 

81479 

59389 

80455 

60784 

79406 

62160 

78333 

635i8 

77236 

34 

11 

57999 
58023 

81462 

59412 

8043  8 

60807 

79388 

62183 

783i5 

63540 

77218 

33 

81445 

59436 

80420 

6o83o 

79371 

62206 

78297 

63563 

77199 

32 

29 

58047 

81428 

59459 

8o4o3 

6o853 

79353 

62229 

78279 

63585 

77181 

3! 

3o 

58070 

81412 

59482 

8o386 

60876 

79335 

6225l 

78261 

636o8 

77162 

3o 

3i 

58094 

81395 

59506 

■8o368 

60899 

79318 

62274 

78243 

63630 

77144 

29 

32 

58ii8 

81378 

59529 

8o35i 

60922 

79300 

62297 

78225 

63653 

77120 

28 

33. 

58i4i 

8i36i 

59552 

8o334 

60945 

79282 

62320 

78206 

63675 

77107 

27 

34 

58 1 65 

8i344 

59376 

8o3i6 

60968 

79264 

62342 

78188 

63698 

77088 

26 

35 

58189 

8i327 

59599 

80299 

60991 

79247 

62365 

78170 

63720 

77070 

23 

36 

58212 

8i3io 

59622 

80282 

6ioi5 

79229 

62388 

78152 

63742 

77o5i 

24 

37 

58236 

81293 

59646 

80264 

6io38 

79211 

62411 

78134 

63765 

77033 

23 

38 

5S260 

81276 

59669 

80247 

61061 

79193 

62433 

78116 

63787 

77014 

22 

39 

58283 

81259 

59693 

8o23o 

61084 

79176 

62456 

78098 

638 10 

76996 

21 

40 

58307 

81242 

59716 

80212 

61107 

79158 

62479 

78079 

63832 

76977 

20 

41 

5S33o 

81225 

39739 
59763 

80193 

6ii3o 

79140 

62502 

78061 

63854 

76959 

19 

42 

58354 

81208 

80178 

6ii53 

79122 

62524 

78043 

63877 

76940 

18 

43 

58378 

81191 

D97S6 

80160 

61176 

79103 

62547 

78023 

63899 

76921 

17 

44 

58401 

81174 

59S09 

80143 

61199 

79087 

62570 

78007 

63922 

76903 

76884 

16 

45 

58425 

81157 

59832 

80125 

61222 

79069 

62592 

77988 

63944 

i5 

46 

58449 

81140 

59836 

80108 

61245 

7905 1 

62615 

77970 

63966 

76866 

i4 

47 

58472 

81123 

59S79 

80091 

61268 

79033 

62638 

77952 

63989 

76847 

i3 

48 

58496 

8uo6 

59902 

80073 

61291 

79016 

62660 

77934 

64011 

76828 

12 

49 

585i9 

81089 

59926 

8oo56 

6i3i4 

78998 

62683 

77016 
77897 

64o33 

76810 

11 

5o 

58543 

81072 

59949 

8oo38 

61337 

78980 

62706 

64o56 

76791 

10 

5i 

58567 

8io55 

59972 

80021 

6i36o 

78962 

62728 

77879! 

64078 

76772 

? 

8 

52 

58590 

8io38 

39995 

8ooo3 

6i383 

78944 

62751 

77861 

64100 

76704 

53 

58614 

81021 

60019 

79986 

61406 

78926 

62774 

77843 

64123 

76730 

7 

54 

58637 

81004 

60042 

79968 

61429 

78908 
78891 

62796 

77824 

64145 

76717 

6 

55 

58661 

80987 

6oo65 

79951 

61401 

62819 

77806 

64167 

76698 

5 

56 

53684 

80970 

60089 

79934 

61474 

78873 

62842 

77788 

64190 

76679 

4 

57 

58708 

80953 

60112 

79916 

61497 

78855 

62864 

77769 

64212 

76661 

3 

58 

IM' 

80936 

60 1 35 

79899 

6i52o 

78837 

62887 

7775i 

64234 

76642 

3 

59 

58755 

80919 

601 58 

79S81 

61 543 

78819 

62909 

77733 

64256 

76623 

I 

60 

58779 

80902 

60182 

79864 

61 566 

78801 

62932 

77715 

64279 

76604 

0 

/ 

Cosine 

Sine. 

1  Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

54° 

53° 

52=" 

51° 

60° 

f 

1-J 


72           NATURAL  SIXES  AND  COSINES.     .  Table  III. 

t 

40° 

41° 

42° 

43°    1 

44° 

f 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

0 

64279 

76604 

656o6 

75471 

66913 

743 14 

68200 

73i35 

69466 

71904 

60 

1 

64301 

76586 

65628 

75452 

66935 

74295 

68221 

73116 

69487 

71914 

g 

2 

64323 

76567 

65650 

75433 

66956 

74276 

68242 

73096 

69508 

71894 

3 

64346 

76548 

65672 

75414 

66978 

74256 

68264 

73016 

69529 

71873 

57 

4 

64368 

76530 

65694 

75395 

66999 

74237 

68285 

73o56 

69549 

71853 

56 

5 

64390 

7651 1 

63716 

75373 

67021 

74217 

683o6 

73o36 

69570 

71833 

55 

6 

64412 

76492 

63738 

75356 

67043 

74198 

68327 

73016 

69591 

71813 

54 

7 

64435 

76473 

65739 

75337 

67064 

74178 

68349 

72996 

69612 

71792 

53 

8 

64457 

76455 

65781 

75318 

67086 

74139 

68370 

72976 

69633 

71772 

52 

9 

64479 

76436 

658o3 

75209 

67107 

74139 

68391 

72937 

69654 

71752 

5i 

10 

64501 

76417 

65825 

75280 

67129 

74120 

68412 

72937 

69675 

71732 

5o 

II 

64324 

76398 

65847 

75261 

67151 

74 1 00 

68434 

72917 

69696 

71711 

49 

12 

64546 

76380 

65869 

73241 

67172 

74080 

68455 

72897 

69717 

71691 

48 

i3 

64568 

76361 

65891 

75222 

67194 

■74061 

68476 

72877 

69737 

71671 

47 

14 

64390 

76342 

6591 3 

75203 

67215 

74041 

68497 

72837 

69758 

7i65o 

46 

13 

64612 

76323 

65935 

75184 

67237 

74022 

68318 

72837 

69779 

7i63o 

43 

i6 

64635 

763o4 

65956 

75i65 

67258 

74002 

68539 

72817 

69800 

71610 

44 

17 

64657 

76286 

65978 

75146 

67280 

73983 

68561 

72797 

69821 

71590 

43 

18 

64679 

76267 

66000 

75126 

67301 

73963 

68582 

72777 

69842 

71569 

42 

19 

64701 

76248 

66022 

75107 

67323 

73944 

686o3 

72757 

69862 

71549 

41 

20 

64723 

76229 

66044 

75088 

67344 

73924 

68624 

72737 

69883 

71529 

40 

21 

64746 

76210 

66066 

75069 

67366 

73go4 

68645 

72717 

69904 

7i5o8 

39 

22 

64768 

76192 

66088 

75o5o 

67387 

73885 

68666 

72697 

69925 

71488 

38 

23 

64790 

76173 

66109 

75o3o 

67409 

73865 

68688 

72677 

69946 

71468 

37 

24 

64812 

76134 

66i3i 

75011 

67430 

73846 

68709 

72657 

69966 

71447 

36 

25 

64834 

76135 

66 1 53 

74992 

67452 

73826 

68730 

72637 

69987 

71427 

35 

26 

64856 

76116 

66173 

74973 

67473 

73806 

68751 

72617 

70008 

71407 

34 

27 

64878 

76097 

66197 

74953 

67495 

73787 

68772 

72597 

70029 

71386 

33 

28 

64901 

76078 

66218 

74934 

67516 

73767 

68793 

72577 

70049 

71366 

32 

29 

64923 

76059 

66240 

74913 

67538 

73747 

68814 

72557 

70070 

71345;  3i  | 

3o 

64945 

76041 

66262 

74896 

67559 

73728 

68835 

72537 

70091 

7i325 

3o 

3i 

64967 

76022 

66284 

74876 

67580 

73708 

^fo^^ 

72517 

70112 

7i3o5 

30 

32 

64989 

76003 

663o6 

74857 

67602 

73688 

^fo^^ 

72497 

70i32 

71284 

28 

33 

65ou 

75984 

66327 

74838 

67623 

73669 

68899 

72477 

70153 

71264 

27 

34 

65o33 

73965 

66349 

74818 

67645 

73649 

68920 

72457 

70174 

71243 

26 

35 

65o55 

75946 

66371 

74799 

67666 

73629 

68941 

72437 

70195 

71223 

23 

36 

65o77 

73927 

66393 

74780 

67688 

73610 

68962 

724:7 

70215 

71203 

24 

37 

65 1  DC 

73908 

66414 

74760 

67709 

73590 

68983 

72397 

70236 

71182 

23 

38 

65l22 

75S89 

66436 

74741 

67730 

73570 

69004 

72377 

70257 

71162 

22 

39 

63144 

73870 

66438 

74722 

67752 

73331 

69025 

72337 

70277 

71141 

21 

40 

65i66 

7585i 

66480 

74703 

67773 

73531 

69046 

72337 

70298 

71121 

20 

41 

65i88 

75832 

66501 

74683 

67795 

735ii 

69067 

72317 

7o3i9 

71100 

19 
18 

42 

65210 

758i3 

66523 

74664 

67816 

73491 

69088 

72297 

70339 

71080 

43 

65232 

75794 

66545 

74644 

67837 

73472 

69109 

72277 

7o36o 

71059 

17 

44 

65254 

73773 

66566 

74625 

67859 

73452 

69130 

72257 

7o38i 

71039 

16 

45 

65276 

73756 

66588 

74606 

67880 

73432 

69131 

72236 

70401 

71019 

ID 

46 

65298 

75738 

66610' 

74586 

67901 

73413 

69172 

72216 

70422 

70998 

14 

47 

65320 

75719 

66632 

74567 

67923 

73393 

69193 

72196 

70443 

70978 

i3 

48 

65342 

75700 

66653 

74548 

67944 

73373 

69214 

72176 

70463 

70937 

12 

49 

65364 

75680 

66675 

74528 

67965 

73353 

69235 

72i56 

70484 

70937 

II 

5o 

65386 

75661 

66697 

74509 

67987 

73333 

69236 

72i36 

7o5o5 

70916 
70896 

10 

5i 

65408 

75642 

^67 18 

74489 

68008 

73314 

69277 

72116 

70323 

9 

52 

6543o 

75623 

66740 

74470 

68029 

73294 

69298 

72095 

70546 

70875 

0 

53 

65432 

73604 

66762 

74451 

68o5i 

73274 

69319 

72073 

70367 

70855 

7 

54 

65474 

75585 

66783 

74431 

68072 

73254 

69340 

72055 

70587 

70834 

6 

55 

65496 

75566 

668o5 

74412 

68093 

73234 

69361 

72035 

70608 

70813 

5 

56 

655i8 

75547 

66827 

74392 

68ii5 

73215 

69382 

72015 

70628 

70793 

4 

^7 

65540 

75528 

66848. 

74373 

68i36 

73195 

69403 

71995 

70649 

70772 

3 

58 

65562 

75509 

66870 

74353 

68157 

73175 

69424 

71974 

70670 

70732 

3 

59 

65584 

75490 

66891 

?4334 

68179 

73i55 

69445 

71954 

70690 

70731 

I 

60 

656o6 

75471 

66913 

743 1 4 

68200 

73i35 

69466 

71934 

707 1 1 

707  n 

'0 

/ 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

1 

49° 

48° 

47° 

46° 

46° 

Tadi,e  III.     -fiATURAL  TANGENTS  AND  COTANGENTS.     13  | 

/ 

0 

0° 

1° 

2° 

J 

° 

60 

Tangent. 

Cotang. 

Tangent.  Cotang. 

Tangent. 

Cotang. 

Tange.it. 

Cotang. 

00000 

Infinite. 

01746 

57  •  2900 

03492 

28^6363 

o524l 

I9^o8ii 

I 

00029 

3437-75 
1718.87 

01775 

56 

3do6 

o352I 

28 

3994 

05270 

18 

0755 

59 

2 

00058 

01804 

55 

44i5 

o355o 

28 

1664 

05290 
o5323 

18 

8711 

58 

3 

00087 

114 

5-92 

01833 

54 

56i3 

03579 

27 

9372 

18 

7678 

57 

4 

00116 

859 

•436 

01S62 

53 

7086 

o36o9 

27 

7117 

05357 

18 

6656 

56 

5 

00I4D 

687 

•549 

01891 

52 

8821 

03638 

27 

4899 
271D 

05387 

18 

5645 

55 

6 

00175 

572 

•  957 

01920 

52 

0807 

03667 

27 

o54i6 

18 

4645 

54 

1 

00204 

491 

•  106 

01940 

5i 

3o32 

03696 

27 

0D66 

05445 

18 

3655 

53 

8 

00233 

.429 

.718 

01978 

Do 

5485 

03725 

26 

8450 

05474 

18 

2677 

52 

9 

00262 

38i 

•971 

02007 

49 

8i57 

03754 

26 

6367 

o55o3 

18 

1708 

5i 

10 

00291 

343 

•774 

02o36 

49 

1039 

03783 

26 

43i6 

o5533 

18 

07D0 
9802 
8863 

5o 

11 

00320 

3l2 

•521 

02066 

48 

4121 

o38i2 

26 

2296 

o5562 

17 

49 

12 

oo349 

286 

•47B 

02095 

47 

7395 

o3842 

26 

o3o7 

05591 

17 

48 

i3 

00378 

264 

•441 

02124 

47 

o8d3 

o387i 

25 

8348 

o5620 

17 

7934 

47 

14 

00407 

245 

•  552 

02 1 53 

46 

4489 

03900 

25 

6418 

02649 
05678 

17 

7015 

46 

ID 

00436 

229 

•  182 

02182 

45 

8294 

03929 

25 

45j7 

17 

6106 

45 

i6 

00465 

214 

•858 

02211 

45 

2261 

03958 

2D 

2644 

05708 

17 

52o5 

44 

17 

00495 

202 

•219 

02240 

44 

6386 

03987 

2D 

0798 

05737 

17 

43i4 

43 

i8 

oo524 

190 

.984 

02260 
02298 

44 

0661 

04016 

24 

8978 

05766 

17 

3432 

42 

'9 

oo553 

180 

•932 

43 

5o8i 

04046 

24 

7185 

05795 

17 

2558 

41 

2C 

oo582 

171 

•885 

02328 

42 

9641 

04075 

24 

5418 

o5824 

17 

1693 
0837 

40 

21 

oo6ji 

1 63 

•700 

02357 

42 

4335 

04104 

24 

3675 

05854 

17 

39 

22 

00640 

1 56 

•209 

02386 

41 

9i58 

04 1 33 

24 

1957 

05883 

16 

9990 

38 

23 

00669 

149 

•46D 

024i5 

41 

4106 

04162 

24 

0263 

05912 

16 

OlDO 

83i9 

37 

24 

00698 

143 

•237 

02444 

40 

9'74 

04191 

23 

8593 

05941 

16 

36 

25 

00727 

137 

•  5o7 

02473 

40 

4358 

04220 

23 

6945 

05970 

16 

7496 
6681 

35 

26 

00756 

l32 

•219 

02502 

39 

9655 
D059 

o425o 

23 

5321 

05999 

16 

34 

27 

00785 

127 

•321 

0253 1 

39 

04270 

23 

3718 

06029 
o6o58 

16 

5874 

33 

28 

00814 

122 

•774 

0256o 

39 

o568 

043  08 

23 

2137 

16 

507D 

32 

29 

00844 

118 

•540 

02589 

38 

6177 

04337 

23 

0577 

06087 

16 

4283 

3i 

3o 

00873 

114 

•589 

02619 

38 

1 885 

04366 

22 

9o38 

061 16 

16 

3499 

3o 

3i 

00902 

no 

•  892 

02648 

37 

7686 

04395 

22 

7519 

06145 

16 

2722 

29 

32 

00931 

107 

•  426 

02677 

37 

3579 

04424 

22 

6020 

06175 

16 

1952 

28 

33 

00960 

104 

•171 

02706 

36 

9560 

04454 

22 

4541 

06204 

16 

1190 
0435 

27 

34 

00989 

101 

•107 

02735 

36 

D627 

04483 

22 

3o8i 

06233 

16 

26 

33 

01018 

98. 

2179 

02764 

36 

1776 

04312 

22 

1640 

06262 

i5 

0687 
8945 

25 

36 

01047 

95- 

4893 

02793 

35 

8006 

04541 

22 

0217 

06291 

i5 

24 

il 

01076 

92- 

9085 

02822 

35 

43 1 3 

04570 

21 

88i3 

o632i 

i5 

8211 

23 

38 

oiio5 

90- 

4633 

0285i 

35 

0695 

04590 

21 

7426 

o635o 

i5 

7483 

22 

39 

oii3p 

88- 

1436 

02881 

34 

71D1 

04628 

21 

6od6 

06379 

i5 

6762 

21 

40 

01164 

85- 

9398 

02910 

34 

3678 

04658 

21 

4704 

06408 

i5 

6048 

20 

41 

01193 

83- 

8435 

02939 

34 

0273 

04687 

21 

3369 

06437 

i5 

5340 

19 

42 

01222 

81. 

8470 

02968 

33 

6935 

04716 

21 

2049 

06467 

i5 

4638 

18 

43 

0125l 

7Q' 

9434 

02997 

3i 

3662 

04745 

21 

0747 

06496 

i5 

3943 

17 

44 

01 280 

78- 

1263 

o3o26 

33 

0452 

04774 

20 

9460 

o6525 

i5 

3254 

16 

45 

01 309 

76. 

3900 

o3o55 

32 

73o3 

04803 

20 

8188 

06554 

i5 

2571 

i5 

46 

oi338 

74- 

7292 

o3o84 

32 

42i3 

04832 

20 

6932 

06584 

i5 

1893 

j4 

47 

01367 

73- 

1390 

o3ii4 

32 

1181 

04862 

20 

5691 

066 1 3 

i5 

1222 

i3 

48 

01396 

71- 

61D1 

o3i43 

3i 

82o5 

04891 

20 

4465 

06642 

i5 

o557 

12 

49 

01425 

70- 

1 533 

o3i72. 

3i 

5284 

04920 

20 

3253 

06671 

J4 

9898 

II 

DO 

01455 

68- 

75oi 

o320I 

3i 

2416 

04940 

20 

2o56 

06700 

14 

0244 

10 

5i 

01484 

67- 

4019 

o323o 

3o 

9599 

0497» 

20 

0872 

o^T^o 

14 

8596 

t 

D2 

oi5i3 

66- 

io55 

o3259 

3o 

6833 

o5oo7 

19 

9702 

067  :)9 

14 

7934 

53 

oi542 

64- 

S58o 

o3288 

3o 

4116 

o5o3t 

19 

8546 

06788 

14 

7317 

1 

54 

01571 

63- 

5567 

o33i7 

3o 

1446 

o5o66 

'9 

74o3 

06817 

14 

6685 

6 

55 

01600 

62- 

4992 

03346 

29 

8823 

o5o95 

19 

6273 

06847 

14 

6059 

5 

56 

01629 

61- 

3^29 

03376 

29 

6245 

o5i24 

19 

5i56 

06876 

14 

5438 

4 

57 

01 658 

6o- 

3o5S 

o34o5 

29 

3711 

o5i53 

'9 

4o5i 

06905 

14 

4823 

3 

58 

01687 

59- 

2659 

03434 

29 

1220 

o5i82 

19 

2059 
1879 

06934 

14 

4212 

2 

59  ]  01716 

58- 

2612 

03463 

28 

8771 

o5212 

'9 

06963 

14 

3607 

I 

60 

61746 

57 • 2900 

03492 

23^6363 

o524i 

i9^o8ii 

06993 

i4'3oo7 

0 

/ 

Colang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

; 

S'.P 

88° 

87° 

8 

0° 

74     NATURAL  TANGENTS  AND  COTANGENTS.     Tablk  III.  | 

/ 

4 

0 

5° 

6° 

7° 

/ 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

0 

06993 

14 -3007 

08749 

11-4801 

I05l0 

9^5i486 

12278 

8-14435 

60 

I 

07022 

14 

241 1 

08778 

1 1 

-3919 

io54o 

9 

48781 

i23o8 

8 

12481 

59 

2 

07o5l 

I4 

1821 

08807 

II 

-3540 

10569 

9 

46141 

12388 

8 

lo536 

53 

3 

07080 

14 

1235 

0S837 

1  I 

-3i63 

10599 

9 

485i5 

12867 

8 

08600 

57 

4 

07110 

i4 

o655 

08866 

II 

.2789 

10628 

9 

40904 

12897 

8 

06674 

56 

5 

07189 

14 

0079 

08895 

II 

•2417 

10657 

9 

88807 

12426 

8 

04756 

55 

6 

07168 

i3 

n5o7 
8940 

08925 

11 

•  2048 

10687 

9 

83724 

12456 

8 

02848 

54 

7 

07197 

33 

08954 

II 

•  1681 

10716 

9 

88154 

12485 

8 

0094S 

53 

8 

07227 

1 3 

8378 

08983 

II 

-i3i6 

10746 

9 

80599 

I25i5 

7 

99o58 

52 

9 

07256 

i3 

7821 

09013 

II 

-0954 

10775 

9 

28o58 

12544 

7 

97176 

5i 

10 

07285 

i3 

7267 

09042 

II 

-0394 

io8o5 

9 

25530 

12574 

7 

95802 

5o 

II 

07314 

i3 

6719 

0907 1 

11 

•0287 

10884 

9 

28016 

12603 

7 

98488 

49 

12 

07344 

i3 

6174 

09101 

10 

.9882 

jo863 

9 

2o5i6 

12638 

7 

9i582 

48 

i3 

07373 

i3 

5634 

09180 

10 

.9529 

10898 

9 

18028 

12662 

7 

89734 

47 

i4 

07402 

i3 

5098 

091 59 

10 

-Q178 

10922 

9 

i5554 

12692 

7 

87895 

46 

i5 

07431 

i3 

4566 

09189 

10 

-8S29 

10952 

9 

18098 

12722 

7 

86064 

45 

i6 

07461 

i3 

4039 
35i5 

09218 

10 

•  8483 

1 098 1 

9 

10646 

12751 

7 

84242 

44 

17 

07490 

i3 

09247 

10 

■  8189 

IIOII 

9 

082 1 1 

12781 

7 

82428 

43 

i8 

07519 
07548 

i3 

2996 

09277 

10 

•7797 

1 1040 

9 

05789 

12810 

7 

80622 

42 

•9 

i3 

2480 

09806 

10 

•7457 

1 1070 

9 

08879 

12840 

7 

78825 

41 

20 

07578 

1 3 

1969 

09335 

10 

-7119 

1 1 099 

00983 

12869 

7 

77035 

40 

21 

07607 

i3 

1461 

09865 

10 

•6783 

1 1 128 

8 

98598 

12899 

7 

75254 

39 

22 

07636 

i3 

0958 

09894 

10 

■  645o 

iii58 

8 

96227 

12920 

7 

78480 

88 

23 

07665 

i3 

0408 

09428 

10 

-6118 

11187 

8 

98867 

12953 

7 

71715 

37 

24 

07695 

12 

9962 

09453 

10 

-5789 

11217 

8 

91520 

12988 

7 

60957 

36 

25 

07724 

12 

9469 

09482 

10 

•5462 

1 1 246 

8 

89183 

18017 

7 

68208 

35 

26 

07753 

12 

898] 

09311 

10 

■5i36 

11276 

8 

86862 

i3o47 

7 

66466 

34 

27 

07782 

12 

8496 

09341 

10 

■4813 

ii3o5 

8 

84551 

18076 

7 

64732 

33 

28 

07812 

12 

8014 

09370 

10 

•4491 

II385 

8 

82252 

i3io6 

7 

63oo5 

32 

29 

07841 

12 

7536 

09600 

10 

•4172 

11864 

8 

79964 

i3i36 

7 

61287 

3i 

3o 

07870 

12 

7062 

09629 

10 

•3854 

1 1894 

8 

77689 

i8i65 

7 

59575 

3o 

3i 

07899 

12 

6591 

09658 

10 

•3538 

11428 

8 

75425 

18195 

7 

57872 

29 

32 

07920 

12 

6124 

09688 

10 

-3224 

11452 

8 

73172 

l3224 

7 

56176 

28 

33 

07958 

12 

566o 

09717 

10 

-2918 

1 1482 

8 

70981 

i3254 

7 

54487 

27 

34 

07987 

12 

5i99 

09746 

10 

■  2602 

ii5ii 

8 

68701 

18284 

7 

52806 

26 

35 

08017 

12 

4742 

09776 

10 

■2294 

1 1 541 

8 

66482 

i33i3 

7 

5ii32 

25 

36 

08046 

12 

4288 

09805 

10 

•1988 

1:570 

8 

64275 

18843 

7 

49465 

24 

37 

08075 

12 

3838 

00884 

10 

•i683 

1 1 600 

8 

62078 

18072 

7 

47806 

23 

38 

08104 

12 

339a 

09864 

10 

•i38i 

1 1629 

8 

59898 

18402 

7 

46154 

22 

39 

08 1 34 

12 

2946 

09898 

10 

•  1080 

1 1659 

8 

57718 

18482 

7 

44509 

21 

40 

08 1 63 

12 

25o5 

09928 

10 

•0780 

1 1688 

8 

55555 

13461 

7 

42871 

20 

41 

08192 

12 

2067 

09952 

10 

•0483 

11718 

8 

58402 

13491 

7 

41240 

!? 

42 

08221 

12 

i632 

09981 

10 

•0187 

11747 

8 

5i25o 

i352i 

7 

39616 

43 

o825i 

12 

1201 

lOOII 

9- 

58980 

11777 

8 

49128 

i855o 

7 

37999 

\l 

44 

08280 

12 

0772 

10040 

9- 

56007 

11806 

8 

47007 

i358o 

7 

86889 

45 

o83o9 

12 

o346 

10069 

9- 

98101 

1 1 836 

8 

44896 

i36o9 

7 

34786 

i5 

46 

08339 

II 

9923 

10128 

9- 

5021 1 

§7888 

1 1 865 

8 

42795 

13689 

7 

33190 

14 

47 

08368 

II 

9304 

9- 

11893 

8 

40705 

18669 

7 

3 1600 

i3 

48 

08397 

II 

9087 

ioi58 

9- 

B4482 

11924 

8 

38625 

18698 

7 

3ooi8 

12 

49 

08427 

II 

8673 

10187 

9' 

S1641 

11934 

8 

36555 

18728 

7 

28442 

11 

5o 

08456 

II 

8262 

10216 

9- 

78817 

11988 

8 

34496 

18758 

7 

26873 

10 

5i 

08485 

II 

7853 

10246 

9- 

76009 

I20i3 

8 

32446 

13787 

7 

253io 

q 

52 

o85i4 

II 

7448 

10273 

9- 

78217 

12042 

8 

3o4o6 

i38i7 

7 

28754 

8 

53 

08544 

II 

7045 

io3o5 

9- 

70441 

12072 

8 

28876 

18846 

7 

12204 

7 

54 

08573 

II 

6645 

10834 

9- 

57680 

12101 

8 

26855 

13876 

7 

20661 

6 

55 

08602 

II 

6248 

io368 

9- 

54983 

12181 

8 

24345 

10906 

7 

19125 

5 

56 

08632 

II 

5853 

10398 

9' 

52205 

12160 

8 

22344 

18985 

7 

17594 

4 

57 

08661 

II 

5461 

10422 

9- 

5o490 

12190 

8 

20352 

18965 

7 

16071 

3 

58 

08690 

II 

5072 

10452 

9- 

56791 

12219 

8 

1S370 

18995 

7 

14553 

a 

59 

08720 

II 

4685 

1 048 1 

9- 

54106 

12240 

8 

16898 

14024 

7 

i3o4a 

I 

60 

08749 

ii-43oi 

io5io 

9-51436 

12278 

8 

14435 

i4o54 

7-11D37 

0 

t 

Cotang. 

Tangent. 

Cotang.  1  Tangent. 

Cotang. 

T 

angent. 

Cotang. 

Tangent. 

/ 

8 

5° 

84° 

83° 

82° 

Table  III.     NATURAL  TANGENTS  AND  COTANGENTS,     16  | 

1 

8° 

9° 

10° 

11° 

1 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

o 

i4o54 

7-11537 

i5838 

6.31375 

17633 

5-67128 

19438 

5-14455 

60 

I 

14084 

I0038 

i5868 

6 

30189 

17663 

5 

661 65 

19468 

5 

13658 

59 
58 

3 

i4ii3 

08546 

15898 

6 

29007 

17693 

5 

652o5 

19498 

5 

12862 

3 

14143 

07059 

15928 

6 

27829 

17723 

5 

64248 

19029 

5 

12069 

57 

4 

14173 

05579 

15958 

6 

26653 

17753 

5 

63295 

19039 

5 

11279 

56 

5 

14202 

04 1  o.l 

15988 

6 

25486 

17783 

5 

62344 

19589 

0 

1 0490 

55 

6 

14232 

02637 

16017 

6 

24321 

17813 

5 

61.I97 

19619 

5 

00704 
08921 

54 

7 

14262 

01174 

16047 

6 

23i6o 

17843 

5 

60402 

19649 

5 

53 

8 

14291 

6 

99718 

16077 

6 

22003 

17873 

5 

59511 

19680 

5 

08139 

52 

9 

I432I 

6 

98268 

16107 

6 

2o85i 

17903 

5 

58573 

19710 

5 

07360 

5i 

10 

i435i 

6 

96823 

16137 

6 

19703 

17933 

5 

57638 

19740 

5 

06584 

5o 

II 

14381 

6 

95385 

16167 

6 

18559 

17963 

5 

56706 

19770 

5 

o58o9 

49 

48 

12 

14410 

6 

93(^52 

16196 

6 

17419 

17993 

5 

sS^I 

19801 

5 

o5o37 

i3 

1444b 

6 

92023 

16226 

6 

16283 

18023 

5 

19831 

5 

04267 

2 

1 4 

14470 

6 

91104 

16256 

6 

i5i5i 

i8o53 

5 

53927 

19861 

5 

03499 

i5 

14499 

6 

89688 

16286 

6 

14023 

i8o83 

5 

53007 

19891 

5 

02734 

43 

i6 

14529 

6 

88278 

i63i6 

6 

12899 

i8ii3 

5 

52090 

1 992 1 

5 

01971 

44 

17 

14559 

6 

86874 

16346 

6 

11779 

18143 

5 

51176 

19902 

5 

01210 

43 

i8 

14588 

6 

85473 

16376 

6 

10664 

18173 

5 

50264 

19982 

5 

0045 1 

42 

•9 

14618 

6 

84082 

16405 

6 

09552 

18203 

5 

49356 

20012 

4 

99695 

41 

20 

14648 

6 

82694 

16435 

6 

08444 

18233 

5 

4845 1 

20042 

4 

98940 

40 

21 

14678 

6 

8i3i2 

1 6465 

6 

07340 

18263 

5 

47348 

20073 

4 

98188 

39 

22 

14707 

6 

79936 

16495 

6 

06240 

18293 

5 

46648 

2oio3 

4 

97438 

38 

23 

14737 

6 

78D64 

16525 

6 

o5i43 

i8323 

5 

45751 

2oi33 

4 

96690 

37 

24 

14767 

6 

77199 

16555 

6 

o4o5i 

i8353 

5 

44857 

10164 

4 

95945 

36 

23 

14796 

6 

75838 

i6585 

6 

02962 

18383 

5 

43966 

20194 

4 

95201 

35 

26 

14826 

6 

74483 

i66i5 

6 

01878 

18414 

5 

43077 

20224 

4 

94460 

34 

11 

14856 

e 

73i33 

16645 

6 

00797 

18444 

5 

42192 

20254 

4 

93721 

33 

14886 

6 

71789 

16674 

5 

90720 
98646 

18474 

5 

4i3o9 

20285 

4 

92984 

32 

29 

14915 

6 

70450 

16704 

5 

i85o4 

5 

40429 

2o3i5 

4 

92249 

3i 

3o 

14945 

6 

69116 

16734 

5 

97576 

18534 

5 

39502 

20343 

4 

91016 

3o 

3i 

14975 

6 

67787 

16764 

5 

96510 

18064 

5 

38677 

.20376 

4 

90785 

20 

32 

i5oo5 

6 

66463 

16794 

5 

95448 

18594 

5 

37805 

2o4o6 

4 

90o56 
89330 
886o5 

28 

33 

i5o34 

6 

65 1 44 

16824 

5 

94390 

18624 

5 

36936 

20436 

4 

27 

34 

i5o64 

6 

6383 1 

16854 

5 

93335 

18654 

5 

36070 

20466 

4 

26 

35 

15094 

6 

62523 

16884 

5 

92283 

18684 

5 

35206 

20497 

4 

87882 

25 

36 

i5i24 

6 

61219 

16914 

5 

91235 

18714 

5 

34345 

2o527 

4 

87162 

24 

37 

i5i53 

6 

59921 

16944 

5 

90191 
89151 
88114 

18745 

5 

33487 

20557 

4 

86444 

23 

38 

i5i83 

6 

58627 

16974 

5 

18773 

5 

3263i 

2o58S 

4 

85727 

22 

39 

i52i3 

6 

57339 

17004 

5 

i88o5 

5 

31778 

20618 

4 

8ooi3 

21 

40 

1 5243 

6 

56o55 

17033 

5 

87080 

18835 

5 

30928 

20648 

4 

84300 

20 

41 

15272 

6 

54777 

17063 

5 

86o5i 

i8865 

5 

3oo8o 

20679 

4 

83590 

>9 

42 

i53o2 

6 

535o3 

17093 

5 

85o24 

18895 

5 

29235 

20709 

4 

82882 

18 

43 

15332 

6 

52234 

17123 

5 

84001 

18925 

5 

28393 

20739 

4 

82175 

n 

44 

15362 

6 

50970 

17153 

5 

82982 

18900 

5 

27533 

20770 

4 

81471 

16 

45 

1 539 1 

6 

49710 

17183 

5 

81966 

18986 

5 

26713 

20800 

4 

80769 

i5 

46 

i542i 

6 

48456 

17213 

5 

80953 

19016 

5 

25880 

2o83o 

4 

80068 

14 

47 

i545i 

6 

47206 

17243 

5 

79944 
78938 

19046 

5 

25o48 

20861 

4 

79370 

i3 

48 

1 5481 

6 

45961 

17273 

5 

19076 

5 

24218 

20891 

4 

78673 

12 

49 

i55ii 

6 

44720 

17303 

5 

77936 

19106 

5 

23391 

20921 

4 

7797» 

u 

5o 

1 5540 

6 

43484 

17333 

5 

76937 

19136 

0 

22066 

20952 

4 

77286 

10 

5i 

15570 

6 

42253 

17363 

5 

73941 

19166 

5 

21744 

20982 

4 

76595 

9 

52 

i56oo 

6 

41026 

17393 

5 

74949 

19197 

5 

20925 

2ioi3 

4 

75906 

8 

53 

i563o 

6 

39804 

17423 

5 

73960 

19227 

5 

20107 

21043 

4 

,5219 

7 

54 

i566o 

6 

38587 

17453 

5 

72974 

19207 

^ 

19293 

21073 

4 

74334 

6 

55 

15689 

6 

37374 

17483 

5 

71992 

19287 

5 

18480 

21104 

4 

73851 

5 

56 

15719 

6 

36i65 

I75i3 

5 

71013 

19317 

5 

17671 

2II34 

4 

73170 

4 

57 

15749 

6 

34961 

17543 

5 

70037 

19347 

5 

16863 

21164 

4 

72490 

3 

53 

15779 

6 

33761 

17573 

5 

69064 

19378 

5 

i6o58 

21190 

4 

7i8i3 

2 

59 

1 5809 

6 

32566 

17603 

5 

68094 

19408 

5 

i5256 

21220 

4 

71137 

1 

6o 

15838 

6.31375 

17633 
Cotang. 

5-67128 

19438 

5-14455 

21206 

4-70463 

0 

/ 

C'otang. 

Tangent. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

t 

81° 

80° 

1      '^ 

78° 

76 

NATURAL  TANGENTS  AND  COTANGENTS.     Table  III.  | 

/ 

12° 

■ 

13° 

14° 

15° 

/ 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

0 

21256 

4-70463 

23087 

4-33143 

24933 

4-01078 

26795 

3 -73205 

60 

1 

21286 

4-69791 

23ii7 

4- 

32373 

24964 

4- 

oo582 

26826 

3 

72771 

59 

2 

2i3i6 

4-69121 

23 148 

4- 

32001 

24995 

4- 

00086 

26837 

3 

72338 

58 

3 

21347 

4-68452  1 

23i79 

4- 

3 1 43o 

25o26 

3- 

99392 

26888 

3 

71907 

57 

4 

21377 

4-67786 

23209 

4 

3o86o 

23056 

3 

99099 

26920 

3 

71476 

56 

5 

2!  408 

4-67121 

23240 

4 

30291 

25087 

3 

98607 

26g5i 

3 

71046 

55 

6 

21438 

4-66453 

23271 

4 

29724 

25ii8 

3 

98117 

26982 

3 

70616 

54 

7 

21469 

4-65797 

233oi 

4 

20159 
23593 
28032 

25i49 

3 

97627 

27013 

3 

701S8 

53 

8 

21499 

4-65i38 

23332 

4 

25i8o 

3 

97139 

.  27044 

3 

69761 

52 

9 

21329 

4-64480 

23363 

4 

25211 

3 

96651 

27076 

3 

6o335 
68909 

5i 

\o 

2i56o 

4-63825 

23393 

4 

27471 

23242 

3 

96165 

27107 

3 

5o 

II 

21590 

4-63171 

23424 

4 

2691 1 
26352 

25273 

3 

9563o 

27 1 38 

3 

68485 

49 

48 

12 

2I62I 

4-625i8 

23455 

4 

253o4 

3 

95196 

27169 

3 

68061 

i3 

2i65i 

4-6i868 

23485 

4 

25795 

23335 

3 

94713 

27201 

3 

67633 

47 

i4 

21682 

4-61219 

235i6 

4 

25239 
24685 

25366 

3 

94232 

27232 

3 

67217 

46 

i5 

21712 

4-60572 

23547 

4 

25397 

3 

98751 

27263 

3 

66796 

45 

i6 

21743 

4-59927 

23578 

4 

24l32 

25428 

3 

93271 

27294 

3 

66376 

44 

17 

21773 

4-59283 

236o8 

4 

2358o 

25459 

3 

92793 

27326 

3 

65957 

43 

i8 

21804 

4-58641 

23639 

4 

23o3o 

25490 

3 

92316 

27357 

3 

65538 

42 

19 

21834 

4-58001 

23670 

4 

22481 

25521 

3 

91839 

27383 

3 

65i2i 

4i 

20 

21864 

4-57363 

23700 

4 

21933 

25552 

3 

9 1 364 

27419 

3 

64705 

40 

21 

2 1 895 

4-56726 

23731 

4 

21387 

25583 

3 

90890 

27451 

3 

64289 

% 

22 

21925 

4-56091 

23762 

4 

20842 

256i4 

3 

90417 

27482 

3 

63874 

23 

21956 

4-55458 

23793 

4 

20298 
19756 

25645 

3 

89945 

27513 

3 

63461 

37 

24 

21986 

4-54826 

23823 

4 

25676 

3 

89474 

27545 

3 

63o48 

36 

23 

22017 

4-54196 

23854 

4 

19215 

25707 
25738 

3 

89004 

27576 

3 

62636 

35 

26 

22047 

4-53568 

23885 

4 

18675 

3 

88536 

27607 

3 

62224 

34 

11 

22078 

4-52941 
4-523i6 

23916 

4 

18137 

25769 

3 

88068 

27638 

3 

61814 

33 

22108 

23946 

4 

17600 

25800 

3 

87601 

27670 

3 

6i4o5 

32 

29 

22139 

4-51693 

23977 

4 

17064 

2583 1 

3 

87136 

27701 

3 

60996 

3i 

3o 

22169 

4-51071 

24008 

4 

i653o 

^5862 

3 

86671 

27732 

3 

6o588 

3o 

3i 

22200 

4 -5045 I 

24039 

4 

15997 

25893 

3 

86208 

27764 

3 

60181 

?2 

32 

2223l 

4-49832 

24069 

4 

i5465 

25924 

3 

85745 

27795 

3 

59775 

33 

22261 

4-49215 

24100 

4 

14934 

25955 

3 

85284 

27826 

3 

59370 

27 

34 

22292 

4-48600 

24i3i 

4 

i44o5 

25986 

3 

84824 

27858 

3 

58966 

26 

35 

22322 

4-47986 
4-47374 

24162 

4 

13877 

26017 

3 

84364 

27889 

3 

58562 

25 

36 

22353 

24193 

4 

i335o 

26048 

3 

83906 

27920 

3 

58 160 

24 

37 

22383 

4-46764 

24223 

4 

12825 

26079 

3 

83449 

27952 

3 

57738 

23 

38 

22414 

4-46155 

24254 

4 

I23oi 

26110 

3 

82992 

27983 

3 

57357 

22 

39 

22444 

4-45548 

24285 

4 

11778 

26141 

3 

82537 

280 1 5 

3 

56957 

21 

40 

22475 

4-44942 

243i6 

4 

11256 

26172 

3 

82083 

28046 

3 

56557 

20 

41 

225o5 

4-44333 

24347 

4 

10736 

26203 

3 

8i63o 

28077 

3 

56 159 

10 

18 

42 

22536 

4-43735 

24377 

4 

10216 

26235 

3 

81177 

28109 

3 

55761 

4i 

22567 

4-43134 

24408 

4 

09699 
09182 

26266 

3 

80726 

28140 

3 

55364 

17 

44 

22597 

4-42534 

24439 

4 

26297 

3 

80276 

28172 

3 

54968 

16 

43 

22628 

4-41936 

24470 

4 

08666 

26328 

3 

79827 

28203 

3 

54573 

13 

46 

22658 

4-4i34o 

245oi 

4 

08 1 52 

26359 

3 

79378 
78931 

28234 

3 

54179 

14 

47 

22689 

4  -  40745 

24532 

4 

07639 

26390 

3 

28266 

3 

53785 

i3 

48 

22719 

4-4oi52 

24362 

4 

07127 

26421 

3 

78485 

28297 

3 

53393 

12 

^9 

22750 

4-39560 

24593 

4 

06616 

26452 

3 

78040 

28329 

3 

53ooi 

II 

5o 

22781 

4-38969 

24624 

4 

06107 

26483 

3 

77595 
77152 

28360 

3 

52609 

10 

5i 

228II 

4-38381 

24655 

4 

05599 

265i5 

3 

28391 

3 

52219 

0 

52 

22842 

4-37793 

24686 

4 

05092 

26546 

3 

•76709 

28423 

3 

51829 

8 

53 

22872 

4-37207 

24717 

4 

04586 

26577 

3 

76268 

28454 

3 

5i44i 

7 

54 

22903 

4-36623 

24747 

4 

04081 

26608 

3 

75828 

28486 

3 

5io53 

6 

55 

22934 

4-36o4o 

24778 

4 

03578 

26639 

3 

75388 

28517 

3 

5o666 

5 

56 

22964 

4-35439 

24809 

4 

-o3o75 

26670 

3 

74950 

28549 

3 

50279 

4 

57 

22995 

4-34879 

24840 

4 

-02574 

26701 

3 

74312 

28580 

3 

49894 

3 

53 

23026 

4-343oo 

24871 

4 

■02074 

26733 

3 

74075 

28612 

3 

49505 

2 

59 

23o56 

4-33723 

24902 

4 

-01576 

26764 

3 

•73640 

28643 

3 

49123 

I 

60 

23087 

4-33148 

24933 

4-01078 

26795 

3.73205 

28675 

3-48741 

0 

/ 

1 

Cotang.  Tangent. 

Cotang.  Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

77° 

76° 

75° 

74° 

Table  III. 

NATURAL  TANGENTS  AND  COTANGENTS.     77  | 

,  1 

16° 

17° 

18° 

19°     1 

/ 
60 

i 

rangent. 

Cotang. 

Tangent.!  Cotang. 

Tangent.  Cotang. 

Tangent. 

Cotang. 

0 

28675 

3-48741 

30573 

3 •27083 

32492 

3.07768 

34433 

2^90421  1 

I 

28706 

3. 

48359 

3  060  5 

3. 

26743 

32524 

3- 

07464 

34465 

2^ 

00147 

u 

2 

28738 

3- 

47977 

3o637 

3^ 

26406 

32556 

3^ 

07160 

3449« 

2^ 

89873 

3 

28769 

3. 

47596 

30669 

3- 

26067 

32588 

3^ 

06857 

34530 

2^ 

89600 

57 

4 

28800 

3- 

47216 

30700 

3. 

25729 

32621 

3^ 

06554 

34563 

2^ 

89327 

56 

5 

28832 

3- 

46837 

30732 

3^ 

25392 
25o55 

32653 

3^ 

06252 

34396 

2 

89055 

55 

6 

28864 

3- 

46458 

30764 

3 

32685 

3 

03930 

34628 

2 

88783 

54 

7 

28895 

3- 

46080 

30796 

3 

24719 

32717 

3 

03649 

34661 

2 

88511 

53 

b 

28927 
28958 

3. 

43703 

30828 

3 

24383 

32749 

3 

o5349 

34693 

2 

88240 

52 

9 

3. 

45327 

3o86o 

3 

24049 

32782 

3 

o5o49 

34726 

2 

87970 

5i 

10 

28990 

3- 

44951 

30891 

3 

23714 

32814 

3 

04749 

34758 

2 

87700 

5o 

II 

29021 

3- 

4437^ 

30923 

3 

23381 

32846 

3 

o445o 

34791 

2 

87430 

a 

12 

29053 

3 

44202 

30955 

3 

23048 

32878 

3 

o4i52 

34824 

2 

87161 

i3 

29084 

3 

43829 

30987 

3 

22715 

32911 

3 

03854 

34856 

2 

86892 

47 

U 

29116 

3 

43456 

31019 

3 

22384 

32943 

3 

o3556 

34889 

2 

86624 

46 

i5 

29147 

3 

43o84 

3io5i 

3 

22053 

32975 

3 

o326o 

34922 

2 

86356 

45 

i6 

29179 

3 

42713 

3io83 

3 

21722 

33007 

3 

O2o63 

34954 

2 

860S9 

44 

'7 

29210 

3 

42343 

3iii5 

3 

21392 

33o4o 

3 

02667 

34987 

2 

85822 

43 

i8 

29242 

3 

41973 

3ii47 

3 

21063 

33072 

3 

02372 

35019 

2 

85555 

42 

'9 

29274 

3 

41604 

31173 

3 

20734 

33 104 

3 

02077 

35o52 

2 

83289 

41 

20 

29305 

3 

41236 

3l2IO 

3 

20406 

33i36 

3 

01783 

35o85 

2 

85o23 

40 

21 

29337 
29368 

3 

40869 

3l242 

3 

20079 

33169 

3 

01489 

35117 

2 

84758 

it 

22 

3 

4o5o2 

31274 

3 

19752 

33201 

3 

01196 

33i5o 

2 

84494 

23 

29400 

3 

40136 

3i3o6 

3 

19426 

33233 

3 

00903 

35i83 

2 

84229 

37 

24 

29432 

3 

39771 

3i338 

3 

19100 

33266 

3 

00611 

35216 

2 

83963 

36 

25 

29463 

3 

39406 

31370 

3 

18775 

33298 

3 

oo3i9 

35248 

2 

83702 

35 

26 

29495 

3 

39042 

3 1402 

3 

1 845 1 

33330 

3 

00028 

35281 

2 

83439 

34 

27 

29526 

3 

38679 

31434 

3 

18127 

33363 

2 

99738 

353i4 

2 

83176 

33 

28 

29338 

3 

383 1 7 

3 1 466 

3 

17804 

33395 

2 

99447 

35346 

2 

82914 

32 

29 

29390 

3 

37953 

31498 

3 

174S1 

33427 

2 

99138 

35379 

2 

82653 

3i 

3o 

29621 

3 

37394 

3i53o 

3 

17159 

33460 

2 

98868 

35412 

2 

82391 

3o 

3i 

29653 

3 

37234 

3 1 562 

3 

16838 

33492 

2 

98580 

35445 

2 

82i3o 

29 

32 

29685 

3 

36875 

3 1594 

3 

i65i7 

33524 

2 

98292 

35477 

2 

81870 

28 

33 

29716 

3 

365i6 

31626 

3 

16197 

33557 

2 

98004 

355io 

2 

81610 

27  ■ 

34 

29748 

3 

36i58 

3i653 

3 

1587-7 

33589 

2 

97717 

35543 

2 

8i35o 

26 

35 

29780 

3 

358oo 

31690 

3 

1 5558 

3362  1 

2 

97430 

35576 

2 

81091 

25 

36 

298 1 1 

3 

35443 

31722 

3 

1 5240 

33654 

2 

97144 

35608 

2 

80833 

24 

37 

29843 

3 

35087 

31754 

3 

14922 

33686 

2 

96858 

35641 

2 

80374 

23 

38 

29875 

3 

34732 

31786 

3 

i46o5 

33718 

2 

96373 

35674 

2 

8o3i6 

22 

39 

29906 

3 

34377 

3i8i8 

3 

14288 

33751 

2 

96288 

35707 

2 

80059 

21 

40 

29938 

3 

34023 

3i85o 

3 

13972 

33783 

2 

96004 

35740 

2 

79802 

20 

41 

29970 

3 

33670 

31882 

3 

13656 

338i6 

2 

93721 

35772 

2 

79545 

19 

42 

3oooi 

3 

33317 
3296^ 

31914 

3 

i334i 

33848 

2 

93437 

358oT 

2 

•79289 

18 

43 

3oo33 

3 

31946 

3 

i3o27 

3388i 

2 

95i55 

35838 

2 

•79033 

17 

44 

3oo65 

3 

32614 

31978 

3 

12713 

33913 

2 

t)4872 

35871 

2 

-78778 

It) 

45 

30097 

3 

32264 

32010 

3 

12400 

33945 

2 

94390 

35904 

2 

•78523 

i5 

46 

30128 

3 

31914 

32042 

3 

12087 

33978 

2 

94309 

35937 

2 

•78269 

14 

47 
48 

3oi6o 

3 

3 1 365 

32074 

3 

11775 

■84010 

2 

94028 

35969 

2 

•78014 

i3 

30192 

3 

3i2i6 

32106 

3 

11464 

34043 

2 

93748 

36oo2 

2 

•77761 

12 

49 

3o224 

3 

3o868 

32139 

3 

iii53 

34075 

2 

93468 

36o35 

2 

•77007 

II 

5o 

3o255 

3 

3o52i 

3217. 

3 

10842 

34108 

2 

93 1 89 

36o68 

2 

•77254 

10 

5i 

30287 

3 

30174 

322o3 

3 

io532 

34140 

2 

92910 

36ioi 

2 

•77002 

p 

52 

3o3i9 

3 

29820 
29483 

32235 

3 

•10223 

34173 

2 

92632 

36i34 

2 

•76750 

8 

53 

3o35i 

3 

32267 

3 

•09914 

342o5 

2 

92354 

36167 

2 

.76498 

7 

54 

3o3S2 

3 

20139 

32299 

3 

•  09606 

34238 

2 

•92076 

36199 

2 

•76247 

6 

55 

30414 

3 

28793 

3233i 

3 

■00298 
■08991 

34270 

2 

•91799 
-91523 

36232 

2 

•73996 

5 

56 

30446 

3 

28432 

32363 

3 

343o3 

2 

36265 

2 

•75746 

4 

57 

30478 

3 

-28109 

32396 

3 

•o8685 

34335 

2 

■91246 

36298 
36331 

2 

•75496 

3 

58 

3o5o9 

3 

•27767 

32428 

3 

■08379 

34368 

2 

•90971 

2 

•73246 

a 

59 

3o54r 

3 

•27426 

32460 

3 

•08073 

34400 

2 

•  90696 

36364 

2 

•74997 

I 

60 

30573  1  3-27085 

32492 

3 •07768 

34433 

2 • 9042 1 

36397 

2-74748 

0 

/ 

Cotang.  Tangent. 

Cotang.  Tangent. 

Cotang.  Tangent. 

Cotang. 

Tangent. 

r 

73° 

72° 

71° 

70° 

-78     NATURAL  TANGENTS  AND  COTANGFNTS. 

Table  ]II.  1 

/ 

20° 

21° 

22° 

23° 

1 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent.  Cotang. 

Tangent. 

Cotang. 

0 

36397 
36430 

2-74748 

38386 

2 -60509 

4o4o3 

2 -47500 

42447 

2^35585 

60 

I 

2 

•74499 

38420 

2 

.60283 

40436 

2 

•47302 

42482 

2 

■35395 

59 
58 

2 

36463 

2 

74251 

38453 

2 

60057 

40470 

2 

•47095 

425i6 

2 

-352o5 

3 

36496 

2 

•74004 

38487 

2 

59831 

4o5o4 

2 

•46888 

4255i 

2 

-35oi5 

57 

4 

36529 

2 

73756 

38520 

2 

59606 

4o538 

2 

•46682 

42585 

2 

■34825 

56 

5 

36562 

2 

73509 

38553 

2 

59381 

40572 

2 

46476 

42619 

2 

■34636 

55 

6 

36595 

2 

73263 

38587 

2 

59156 

40606 

2 

46270 

42654 

2 

■34447 

54 

7 

36628 

2 

73017 

38620 

2 

58932 

40640 

2 

46065 

42688 

2 

•34258 

53 

ti 

3666 1 

2 

72771 

38654 

2 

58708 

40674 

2 

•45860 

42722 

2 

•34069 

52 

9 

36694 

2 

72526 

38687 

2 

58484 

40707 

2 

45655 

42737 

2 

•33881 

5i 

10 

36727 

2 

72281 

38721 

2 

58261 

40741 

2 

•45451 

42791 

2 

33693 

5o 

II 

36760 

2 

72o36 

38754 

2 

58o38 

40775 

2 

45246 

42826 

2 

335o5 

S 

12 

36793 

2 

71792 

38787 

2 

57815 

40809 

2 

43043 

42860 

2 

33317 

i3 

36826 

2 

71548 

38821 

2 

57593 

40843 

2 

44839 

42894 

2 

33i3o 

47 

i4 

36859 

2 

7i3o5 

38854 

2 

57371 

40877 

2 

44636 

42929 

2 

32943 

46 

i5 

36892 

2 

71062 

38888 

2 

57160 

40911 

2 

44433 

42963 

2 

•32756 

45 

i6 

36925 

2 

70819 

38921 

2 

56928 

40945 

2 

4423o 

42998 

2 

32570 

44 

17 

36958 

2 

70377 

38955 

2 

56707 

40979 

2 

44027 

43o32 

2 

32383 

43 

i8 

36991 

2 

70335 

38988 

2 

56487 

4ioi3 

2 

43825 

43067 

2 

32197 

42 

19 

37024 

2 

70094 

39022 

2 

56266 

41047 

2 

43623 

43ioi 

2 

32012 

41 

20 

37057 

2 

69853 

39055 

2 

56046 

41081 

2 

43422 

43i36 

2 

31826 

40 

21 

37090 

2 

69612 

39089 

2 

55827 

4iii5 

2 

43220 

43170 

2 

3i64i 

39 

22 

37124 

2 

69371 

39122 

2 

556o8 

41149 

2 

43019 

a32o5 

2 

31456 

38 

23 

37157 

2 

69131 

39156 

2 

55389 

4ii83 

2 

42819 

43239 

2 

31271 

37 

24 

37190 

2 

68892 

39190 

2 

55170 

41217 

2 

42618 

43274 

2 

3 1086 

36 

25 

37223 

2 

68653 

39223 

2 

54952 

4i25i 

2 

42418 

433o8 

2 

30902 

35 

26 

37256 

2 

68414 

39237 

2 

54734 

41285 

2 

42218 

43343 

2 

30718 

34 

27 

37289 

2 

68175 

39290 

2 

545i6 

4i3i9 

2 

42019 

43378 

2 

3o534 

33 

28 

37322 

2 

67937 

39324 

2 

54209 
54082 

4i353 

2 

41819 

43412 

2 

3o35i 

32 

29 

37355 

2 

67700 

39357 

2 

41387 

2 

41620 

43447 

2 

30167 

3i 

3o 

37388 

2 

67462 

39391 

2 

53865 

4I42I 

2 

41421 

43481 

2 

29984 

3o 

3i 

37422 

2 

67225 

39425 

2 

53648 

41455 

2 

41223 

435i6 

2 

29801 

29 

32 

37455 

2 

66989 

39458 

2 

53432 

41490 

2 

41025 

4355o 

2 

29619 

28 

33 

37488 

2 

66752 

39492 

2 

53217 

4i524 

2 

40827 

43585 

2 

29437 

27 

34 

37521 

2 

665i6 

39526 

2 

53ooi 

4i558 

2 

40629 

43620 

2 

29254 

26 

35 

37554 

2 

66281 

39559 

2 

52786 

41592 

2 

40432 

43654 

2 

29073 

25 

36 

37588 

2 

66046 

39593 

2 

52571 

41626 

2 

40235 

43689 

2 

28891 

24 

^ 

37621 

2 

658ii 

39626 

2 

52357 

41660 

2 

4oo38 

43724 

2 

28710 

23 

38 

37654 

2 

65576 

39660 

2 

52142 

41694 

2 

39841 

43758 

2 

28528 

22 

39 

37687 

2 

65342 

39694 

2 

51929 

41728 

2 

39645 

43793 

2 

28348 

21 

40 

37720 

2 

65 1 09 

39727 

2 

5i7i5 

41763 

2 

39449 

43828 

2 

28167 

20 

41 

37754 

2 

64875 

39761 

2 

5i5o2 

41797 

2 

39253 

43862 

2 

27987 

•9 

42 

37787 

2^ 

64642 

39795 

2 

51289 

4i83i 

2 

39058 

43897 

2 

27806 

18 

43 

37820 

2 

64410 

39829 

2 

51076 

41 865 

2 

38862 

43932 

2 

27626 

17 

44 

37853 

2 

64177 

3986^ 

2 

5o864 

41899 

2 

38668 

43966 

2 

27447 

10 

45 

37887 

2 

63945 

39896 

2 

5o652 

41933 

2 

38473 

44001 

2 

27267 

i5 

46 

37920 

2 

63714 

39980 

2 

5o44o 

41968 

2 

38279 

44o36 

2 

27088 

14 

47 

37953 

2 

63483 

39963 

2 

50229 

42002 

2 

38084 

44071 

2 

26909 

i3 

48 

37986 

2 

63252 

39997 

2 

5ooi8 

42o36 

2 

37891 

44io5 

2 

26730 

12 

49 

38o20 

2 

63o2i 

4oo3i 

2 

49807 

42070 

2 

37697 

44140 

2 

26552 

II 

5o 

38o53 

2 

62791 

4oo65 

2 

49597 

42io5 

2 

37504 

44175 

2 

26374 

10 

5i 

38o86 

2 

62561 

40098 

2 

49886 

42139 

2 

37311 

44210 

2 

26196 

9 

52 

38i20 

2 

62332 

4oi32 

2 

49'77 

42173 

2 

37118 

44244 

2 

26018 

8 

53 

38i53 

2 

62103 

40166 

2 

48967 

42207 

2 

36925 

44279 

2 

25840 

7 

54 

38i86 

2 

61874 

40200 

2 

48758 

42242 

2 

36733 

443 14 

2 

25663 

6 

55 

38220 

2 

61646 

40234 

2 

48549 

42276 

2 

36541 

44349 

2 

25486 

5 

56 

38253 

2 

61418 

40267 

2 

48340 

423io 

2- 

36349 

44384 

2^ 

25309 

4 

57 

38286 

2 

61 190 

4o3oi 

2- 

48i32 

42345 

2- 

36i58 

4«i8 

.2^ 

25l32 

3 

58 

38320 

2 

60963 

4o335 

2- 

47924 

42379 

2- 

35967 

44453 

2- 

24956 

a 

59 

38353 

2 

60736 

40369 
4o4o3 

2- 

47716 

424i3 

2- 

35776 

•44488 

2^ 

24780 

1 

60 

.38386 

2 -60509 

2-47509  1 

42447 

2-35585  1 

44523 

2 • 24604 

0 

/ 

Cotang. 

Tangent. 

Cotang.  Tangent. 

Cotang.  Tangent. 

Cotang. 

Tangent. 

/ 

60° 

68° 

67° 

66^^     j 

Table  III.     NATURAL  TANGENTS  AND  COTANGENT 

S.     7».| 

1 

0 

24° 

25° 

26° 

27° 



/ 

Tangent. 

Cotang. 

Tangent.  Cotang. 

Tangent.  C 

otang. 

Tangent.  C 

otang. 

44523 

2 ■ 24604 

46631 

2-i445i 

48773   2 

o5d3o 

50953  •  I 

96261 

60 

I 

44558 

2 

24428 

46666 

2 

14288 

48809   2 

04879 

50989   I 

96120 

?3 

2 

44593 

2 

24232 

46702 

2 

I4I25 

48843   2 

04728 

5io26  I 

93979 

3 

44627 

2 

24077 

46737 

2 

13963 

48881   2 

04577 

5io63  1 

90838 

57 

•  4 

44662 

2 

23902 

46772 

2 

i38oi 

48917   2 

04426 

51099  I 

90698 

56 

5 

44697 
44732 

2 

23727 

46808 

2 

1 3639 

48953   2 

04276 

5ii36  I 

95537 

55 

6 

2 

23553 

46843 

2 

i3477 

48989   2 

04125 

51173  I 

90417 

54 

7 

44767 

2 

23378 

46879 

2 

i33i6 

49026   2 

03975 
o3825 

51209  ' 

95277 

53 

8 

44802 

2 

23204 

46914 

2 

i3i54 

49062   2 

5i246  I 

95137 

52 

9 

44837 

2 

23o3o 

46930 

2 

12093 

12832 

49098   2 

03675 

5i283  I 

94997 

5i 

10 

44872 

2 

22837 

46985 

2 

49134   2 

o3326 

5i3i9  I 

94808 

5o 

II 

44907 

2 

22683 

47021 

2 

12671 

49170   2 

03376 

5i356  I 

94718 

8 

12 

44942 

2 

22310 

47006 

2 

125ll 

49206   2 

o3227 

51393  I 
5i43o  I 

94379 

i3 

44977 

2 

22337 

47092 

2 

i235o 

49242   2 

03078 

94440 

47 

14 

43012 

2 

22164 

47128 

2 

12190 
i2o3o- 

49278   2 

02929 

51467  I 

94301 

46 

i5 

43047 

2 

21992 

47163 

2 

493i5  2 

02780 

5i5o3  I 

94162 

40 

i6 

45082 

2 

21819 

47199 
47234 

2 

11871 

49351  2 

0263 1 

5i54o  I 

94023 

44 

17 

45117 

2 

21647 

2 

11711 

49387  2 

02483 

5i577  I 

93885  43 

i8 

45i52 

2 

21473 

47270 

2 

1:552 

4q423  1  2 

02335 

5i6i4  I 

93746  1  42 

•9 

45187 

2 

2i3o4 

473o5 

2 

1 1392 
11233 

49400  1  2 

02187 

5i65l  1 

93608  1  41 

20 

40222 

2 

2Ji33 

47341 

2 

49493  2 

0203<) 

5i688  I 

93470  '  40 

21 

45207 

2 

20961 

47377 

2 

11075 

49532  !  2 

01891 

51724  I 

93332  39 

22 

43292 

2 

20790 

47412 

2 

10916 

49568   2 

01743 

51761  I 

93196]  38 

23 

45327 

2 

20619 

47448 

2 

10758 

49604   2 

01596 

51798  I 
5i835  I 

93057 

37 

24 

45362 

2 

20449 

47483 

2 

10600 

49640   2 

01449 

92920 

36 

2J 

45397 

2 

20278 

47319 

2 

10442 

49677   2 

oi3o2 

51872  I 

92782 

35 

26 

43432 

2 

20108 

47555 

2 

10284 

49713   2 

on  55 

51909  I 

92645 

34 

;j 

45467 

2 

19938 

47390 

2 

10126 

49749   2 

01008 

51946  I 

925o8 

33 

455o2 

2 

19769 

47626 

2 

09069 

49786   2 

00S62 

51983  I 

92371 

32 

29 

45537 

2 

19399 
19430 

47662 

2 

09811 

49822   2 

00715 

52020   I 

92235 

3i 

3o 

45573 

2 

47698 

2 

09654 

49S58   2 

00069 

52057   I 

92098 

3o 

3i 

45608 

2 

19261 

47733 

2 

09498 

49894   2 

00423 

52094   I 

91962 

^2 

32 

43643 

2 

19092 

47769 

2 

09341 

49931   2 

00277 

52i3i  I 

91826 

28 

33 

45678 

2 

18923 

47805 

2 

09184 

49967   2 

ooi3i 

52168  I 

91690 

27 

34 

45713 

2 

18755 

47840 

2 

09028 

5ooo4  I 

99986 

52200   I 

91504 

26 

35 

45748 

2 

18387 

47876 

2 

08872 

5oo4o  I 

99841 

52242  I 

91418 

20 

36 

45784 

2 

18419 

47912 

2 

08716 

50076  I 

99695 

52279  ' 

91282 

24 

37 

45819 

2 

i825i 

47948 

2 

o856o 

5oii3  I 

99OO0 

523i6  I 

91147 

23 

38 

45854 

2 

18084 

47984 

2 

o84o5 

5oi49  I 
5oi85  I 

99406 

52353  I 

91012 

22 

39 

45889 

2 

17916 

48019 

2 

08250 

99261 

52390  I 

90876 

21 

40 

45924 

2 

17749 

48055 

2 

08094 

5o222   I 

99116 

52427  I 

90741 

20 

4i 

45960 

2 

17582 

4S091 

2 

07939 

5o258  I 

98972 

52464  I 

90607 

'2 

42 

45995 

2 

17416 

48127 

2 

07780 

50295  I 

98828 

525oi  I 

90472 

18 

43 

46o3o 

2 

17249 

48163 

2 

07630 

5o33i   I 

9S684 

52538  I 

90337 

17 

44 

46065 

2 

17083 

48198 
48234 

2 

07476 

5o368  I 

98540 

52575  I 

90203 

16 

45 

46101 

2 

16917 

2 

07321 

5o4o4  I 

98396 

526i3  1 

90069 

i5 

46 

46 1 36 

2 

16751 

48270 

2 

07167 

5o44i  I 

98253 

5265o  I 

89935 

14 

47 

46171 

2 

i6585 

483o6 

2 

07014 

30477  I 

98110 

52687  I 

89801 

i3 

48 

46206 

2 

16420 

48342 

2 

06860 

5o5i4  I 

97066 
97823 

52724  I 

89667 

12 

49 

46242 

2 

16255 

48378 

2 

06706 

5o55o  I 

52761  I 

89533 

II 

5o 

46277 

2 

16090 

48414 

2 

o6553 

5o587  I 

97680 

52708  I 

89400 

10 

5i 

46312 

2 

15925 

48450 

2 

06400 

5o623  I 

97338 

52836  I 

89266 

I 

52 

46348 

2 

13760 

48486 

2 

06247 

5o66o  I 

97395 

52873  I 

89133 

53 

46383 

2 

15596 

48321 

2 

06094 

50696  I 
50733  I 

97203 

52910  I 

89000 

7 

54 

46418 

2 

15432 

48557 

2 

05942 

97111 

52947  I 

88867 

6 

55 

46454 

2 

1 5268 

48593 

2 

05790 

50769  I 

96969 

52984  I 

88734 

5 

56 

46489 

2 

i5io4 

48629 

2 

o5637 

5o8o6  I 

96827 

53022   I 

88602 

4 

57 
58 

46525 

2 

14940 

48663  1  2 

o5485 

50843  I 

96685 

53o59  1 

88469   3 

4656o 

2 

14777 

48701 

2 

05333 

50879  I 

96544 

53096  1 
53i34  I 

88337   2 

59 

46595 
4663 1 

2 

14614- 

48737 

2 

o5i82 

50916  I 

96402 

88205   I 

60 

2 -14451 

48773 

2 

o5o3o 

50953  I 

96261 

53171  I 

88073 1  0 

/ 

Cotang.  Tangent. 

Cotang.  1  T 

dngent. 

Cotang.  T 

angent. 

Cotang.  T 

angent. 

/ 

65^ 

04° 

6.3° 

62° 

80     NATURAL  TANGENTS  AND  COTANGENTS. 

T-VBI.K  nij 

/ 

28° 

29° 

30° 

s 

p 

/ 

Tangent.  C 

/Otang. 

Tangent.  C 

/Otang. 

Tangent.  C 

/Otang. 

Tangent. 

Cotang. 

0 

53 171   l' 

88073 

5543 1   I 

8o4o5 

57735   I 

.73205 

60086 

1-66423 

60 

I 

53208   1 

•87941 

55469   I 

80281 

57774   I 

•73089 

60126 

I -66318 

59 

2 

53246  I 

.87809 

55507   I 

80 1 58 

57813   I 

•72973 

601 65 

1-66209 

5& 

3 

53283  I 

87677 

55545   I 

8oo34 

5785i  I 

•72857 

6o2o5 

1 • 66099 

57 

4 

53320  1 

87546 

55583  I 

799" 

57890  I 

72741 

60245 

1^65990 

56 

5 

53358  I 

87415 

55621  1 

79788 

57920  I 

•72625 

60284 

i^6588i 

55 

6 

53395  I 

87283 

55659  I 

79665 

57968  I 

•72309 

6o324 

1.65772 

54 

7 

53432  I 

87152 

55697  I 

79542 

58007  I 

•72393 

6o364 

1.65603 

53 

8 

53470  I 

87021' 

55736  I 

79419 

58046  I 

72278 

60403 

1.65554 

52 

9 

53507  I 

86891 

55774  I 

79296 

58o85  I 

72163 

60443 

1^65445 

5i 

10 

53545  I 

86760 

558i2  I 

79174 

58124  I 

72047 

60483 

1.65337 

5o 

II 

53582  I 

86630 

5585o  I 

79o5i 

58162  I 

71932 

6o322 

1.65228 

49 

12 

53620  I 

86499 

55888  I 

78929 

58201  I 

71817 

6o562 

I.65I20 

48 

i3 

53657  I 

86309 

55926  I 

78^07 

58240  I 

71702 

60602 

1 -6501 1 

47 

i4 

53694  I 

86239 

55964  I 

78685 

58279  I 

71 588 

60642 

1-64903 

46 

i5 

53732  1 

86109 

56oo3  1 

78563 

583i8  1 

71473 

60681 

1-64795 

45 

i6 

53769  I 

85979 

56o4i  I 

78441 

58357  I 

7i358 

60721 

1-64687 

44 

17 

53807  I 

85850 

56079  ' 

783.0 

58396  I 

71244 

60761 

1-64579 

43 

i8 

53844  I 

85720 

56117  I 

7819S 

58435  I 

71129 

60801 

1-64471 

42 

'9 

53882  I 

85591 

56i56  I 

78077 

58474  I 

71015 

60841 

1-64363 

41 

20 

53920  I 

85462 

56194  I 

77955 

585i3  1 

70901 

60881 

1-64256 

40 

21 

53957  I 

85333 

56232  1 

77834 

58552  1 

70787 

60921 

1-64148 

39 

22 

53995  I 

85204 

56270  1 

77713 

58591  I 

70673 

60960 

I -64041 

38 

23 

54o32  I 

85075 

563o9  I 

77392 

5863 1  I 

7o56o 

61000 

1-63934 

37 

2i 

54070  I 

84946 

56347  I 

77471 

58670  1 

70446 

61040 

I •63826 

36 

25 

54107  I 

84818 

56385  I 

77351 

58709  I 

7o332 

61080 

1^637 1 9 

35 

26 

54145  I 

84689 

56424  I 

77230 

58748  I 

70219 

6II20 

I -63612 

34 

27 
28 

54i83  I 

84561 

56462  1 

77110 

58787  1 

70106 

6II60 

i-635o5 

33 

54220  I 

84433 

565oo  I 

76990 

58826  1 

69992 

61200 

1-63398 

32 

29 

54258  I 

843o5 

56539  I 

76869 

58865  I 

69879 

61240 

1-632Q2 

3i 

3o 

54296  I 

84177 

56577  I 

76749 

58904  1 

69766 

61280 

i-63i85 

3o 

3i 

54333  I 

84049 

566i6  I 

76630 

58944  I 

69653 

6i320 

i-63o79 

29 

32 

54371  I 

83922 

56654  I 

76510 

58983  I 

69541 

6i36o 

1-62972 

28 

33 

54409  I 

83794 

56693  I 

76390 

59022  I 

69428 

61400 

1-62866 

27 

34 

54446  I 

83667 

56731  I 

76271 

59061  I 

69316 

61440 

1-62760 

26 

35 

54484  I 

83540 

56769  I 

76i5i 

59101  I 

69203 

61480 

1-62654 

25 

36 

54522  I 

834 1 3 

56808  1 

76032 

59140  I 

69091 

6i52o 

1-62548 

24 

37 

54560  I 

83286 

56846  I 

73913 

59179  I 

68979 

6i56i 

1-62442 

23 

33 

54397  I 

83 1 59 
83o33 

56885  I 

73794 

59218  I 

68866 

61601 

1-62336 

22 

39 

54635  I 

56923  I 

73675 

59238  1 

68754 

61641 

1-62230 

21 

40 

54673  I 

82906 

56962  I 

73556 

59297  1 

68643 

61681 

1-62125 

20 

41 

54711  I 

82780 

57000  I 

73437 

59336  1 

68531 

61721 

1-62019 

10 

42 

54748  I 

82654 

57039  I 

73319 

59376  I 

68419 

61761 

1-61914 

18 

43 

54786  I 

82528 

57078  I 

75200 

59415  I 

683o8 

61801 

1-61808 

17 

44 

54824  I 

82402 

57116  I 

75082 

59454  I 

68196 

61842 

1-61703 

16 

45 

54862  I 

82276 

57155  I 

74964 

39494  I 

68o85 

61882 

1-61598 

i5 

46 

54900  I 

82i5o 

57.93  I 

74846 

59533  1 

67974 

61922 

1-61493 

14 

47^ 

54938  I 

82025 

57232  I 

74728 

59373  1 

67863 

61962 

i-6i388 

i3 

48 

54975  I 

81899 

57271  I 

74610 

59612  I 

67752 

62oo3 

1-61283 

12 

49 

55oi3  I 

81774 

57309  I 

74492 

59651  1 

67641 

62043 

1-61179 

11 

5o 

55o5i  I 

81649 

57348  I 

74373 

59691  I 

67530 

62083 

1-61074 

10 

5i 

55089  I 

8i524 

57386  I 

74237 

59730  I 

67419 

62124 

1-60970 

9 

52 

55127  I 

81399 

57425  I 

74140 

59770  1 

67309 

62164 

i-6o865 

8 

53 

55i65  I 

81274 

57464  1 

74022 

59809  1 

67198 

62204 

1-60761 

7 

54 

55203  i' 

8ii5o 

57503  1 

73905 

59849  I 

67088 

62245 

1-60657 

6 

55 

55241  I 

8io25 

57541  1 

73788 

59888  I 

66978 

62285 

i-6o553 

5 

56 

55279  I 

S0901 

57080  1 

73671 

59928  I 

66867 

62323 

1-60449 

4 

57 

55317  I 

80777 

57619  I 

73555 

59967  1 

66757 

62366 

i-6o343 

3 

58 

55355  1 

8o653 

57657  I 

73438 

60007  •  • 

66647 

62406 

1-60241 

a 

59 

55393  1 

80529 

57696  1 

73321 

60046  1 

66538 

62446 

1-60137 

1 

60 

55431  1. 

8o4o5 

57735  1 

73205 

60086  I  • 

66428 

62487 

1 -6oo33 

0 

/ 

Cotang.  T 

mgent. 

Cotang.  1  T 

mgent. 

Cotang.  T 

mgent. 

Cotang 

Tangent. 

/ 

01° 

60° 

59° 

5 

30 

Table  IIL 

NATURAL  TANGENTS  AND  COTANG KNTS.     8]  | 

r 

32° 

33° 

34° 

35° 

/ 

Tangent.  C 

'otang. 

Tangent.  Cotang. 

Tangent. 

Cotang. 

Tar  gent.  1  Cotang. 

0 

62487   I 

■6oo33 

64941   I 

•53986 

67451 

I • 482 56 

7C021  1  1 

42815 

60 

I 

62327   1 

■  59930 

649S2   I 

•53888 

67493 
67536 

I -48163 

70064  1  I 

■42726 

59 

2 

62568  I 

.59826 

65o23  I 

•53791 

1-48070 

70107   1 

•42638 

58 

3 

62608  I 

•59723 

65o65  I 

•53693 

67578 

1-47077 
I •47885 

7oi5i  1 

-42330 

57 

4 

62649  1 

.59620 

65io6  I 

•53595 

67620 

70194  I 

-42462  1  56  1 

5 

626S9  I 

.59517 

65i43  I 

•53497 

67663 

1-47792 

70238  1 

•42374!  55  1 

6 

62730  I 

•59414 

65189  I 

•  534(.'0 

67705 

1-47699 

70281  I 

-42286 

54 

7 

62770  I 

593 1 1 

65231  1 

•  533<.2 

67748 

••47607 

7o325  I 

•42198 

53 

8 

6281 1  I 

59208 

65272  I 

•53205 

67790 

I-475I4 

70368  I 

-42110 

52 

9 

62852  I 

59105 

653i4  I 

•53 107 

67832 

1-47422 

70412  1 

-42022 

5i 

10 

62892  I 
62933  I 

59002 

65355  I 

•53010 

67875 

1^47330 

70455  1 

•41934 

5o 

II 

58900 

65397  > 
6543^  1 

•52oi3 

67917 

1-47238 

70499  1  I 

•41847 

49 

12 

62973  I 

5S797 

•52816 

67960 

1-47146 

70542  1  1 

•41759 

48 

i3 

63oi4  I 

58695 

65480  I 

•52719 

68002 

1-47053 

70586  I 

-41672 

47 

i4 

63o55  I 

58593 

6552!  I 

•52622 

68045 

1-46062 

70629  I 

4 1 584 

46 

i5 

63095  I 

58490 

65563  I 

•52525 

68088 

1-46870 

70673  1 

•4i497 

45 

i6 

63i36  I 

58388 

65604  I 

52429 

68i3o 

1-46778 

70717  '  I 

-41409 

44 

17 

63177  J 

58286 

65646  I 

52332 

68173 

1-46686 

70760  1  1 

4l322 

43 

i8 

63217  I 

58 1 84 

65688  1 

52235 

68215 

1-46595 

70804  I 

41235 

42 

19 

63258  I 

58o83 

65729  I 

•52i39 

68258 

i-465o3 

70848  I 

-41148 

41 

20 

63299  I 

57981 

65771  I 

52043 

68301 

: -4641 1 

70891  I 
70935  I 

-41061 

40 

21 

63340  I 

57879 

658i3  I 

51046 

68343 

1-46320 

40974 

39 

22 

63380  I 

57778 

65854  1 

5i85o 

68386 

1-46229 

70979  I 

40887 

38 

23 

63421  I 

57676 

65896  1 
65938  1 

51754 

68429 

1-46137 

71023  I 

40800 

37 

24 

63462  I 

57575 

5 1 658 

68471 

1-46046 

7 1 066  I 

40714 

36 

25 

635o3  I 

57474 

65980  I 

5i562 

685 14 

1-45955 

71110  I 

40627 

35 

26 

63544  1 

57372 

66021  I 

5 1466 

98557 

1-45864 

71134  I 

4o34o 

34 

2^^ 

63584  I 

57271 

66o63  I 

5i37o 

68600 

1-45773 

71198  1 

40434 

33 

63625  I 

57170 

66io5  1 

51275 

68642 

1-45682 

71242  I 

40367 

32 

29 

63666  1 

57069 

66147  I 

51179 

68685 

1-45592 

71285  I 

40281 

3i 

3o 

63707  I 

56969 

66189  1 

51084 

68728 

I -45501 

71329  1  1 

-40195 

3o 

3i 

63748  I 

56868 

6623o  I 

50988 

68771 

I -45410 

71373  !  1 

40109 

29 

32 

63789  I 

56767 

66272  I 

50893 

68814 

1-45320 

71417  1  I 

40022 

28 

33 

63830  I 

56667 

663 14  I 

50797 

68857 

1-45229 

71461   I 

39p36 

27 

34 

63871  I 

56566 

66356  1 

50702 

68900 

i-45i39 

7 1  ,'io5  I 

39850 

26 

35 

63912  I 

56466 

66398  I 

50607 

68942 

I  -45049 

71549  I 
7i5q5  I 
71637  I 

39764 

23 

36 

63953  I 

56366 

66440  I 

5o5i2 

68985 

1-44058 

39679 

24 

^7 

63994  I 
64035  I 

56265 

66482  I 

50417 

69028 

1-44S68 

39393 

23 

38 

56i65 

66524  I 

5o322 

69071 

1-44778 

71681   I 

39507 

22 

39 

64076  I 

56o65 

66566  I 

50228 

69114 

1-44688 

71725  I 

39421 

21 

40 

641 17  I 

55966 

66608  1 

5oi33 

69157 

1-44398 

71769  1 
71813  I 

39336 

20 

41 

641 58  1- 

55866 

6665o  1 

5oo38 

69200 

i-445o8 

39250 

19 

42 

64199  I  • 

55766 

66692  I 

49944 

69243 

I -44418 

71857  I 

39165 

18 

43 

64240  I  • 

55666 

66734  I 

49849 

69286 

1-44329 

71901   1 

39079 

17 

44 

64281  I- 

55567 

66776  I 

49755 

69329 

1-44239 

71946  1 

38994 

16 

45 

64322  1- 

55467 

66818  I 

49661 

69372 

I -44149 

71990  1 

38909 

i5 

46 

64363  I  • 

55368 

66860  I 

49566 

69416 

1-44060 

72034  I 

38824 

14 

H 

64404  I  • 

55269 

66902  I 

49472 

69459 

1-43970 

72078  1 

38738 

i3 

48 

64446  I  • 

55170 

66944  I 

49378 

69302 

I -43881 

72122  1 

38653 

12 

49 

64487  I  • 
64528  I • 

55071 

66986  I 

49284 

69545 

1-43792 

72166  1 

38568 

11 

00 

54972 

67028  I 

49190  j 

69588 

1-43703 

72211   1 

38484 

10 

5i 

64569  I • 

54873 

67071  I 

49097 

69631 

1-43614 

72255  1 

38399 

9 

52 

64610  I- 

54774 

67113  I 

49003 

69675 

1-43525 

72299  I 

383 14 

8 

53 

64652  1 • 

54675 

67155  I- 

48909 

69718 

1-43436 

72344-  I 

38229 

7 

54 

64693  I  • 
64734  I- 

54576 

67197   !• 

48816 

69761 

1^43347 

72388  I 

38143 

6 

55 

54478 

67239   I^ 

48722 

69804 

1^43258 

72432  1 

3  8060 

5 

56 

64775  I  • 

54379 

67282   I 

48629 

69847 

1^43169 

72477  1 

37976 

4 

ll 

64817  I- 

54281 

67324   I 

48536 

69891 

i^43o8o 

72321   I 

37891 

3 

64858  I  • 

54183 

67366   I 

48442 

69934 

1^42992 

72565  1 

37807 

2 

59 

64899  I • 

54085 

67409   I 

48349 

69977 

1 •42903 
1-42815 

72610  1- 

37722 

1 

60 
t 

64941  I • 

53986 

67451   I 

48256 

■70021 

72654  I  - 

37638 

0 

Cotang.  T 

ingent. 

Cotang.  1  T 

ingent. 

Cc<ang. 

Tangent. 

Cotang.  T 

ingent. 

1 

67° 

56°      1 

1 

5 

5°      i      54° 

S2     NATURAL  TANGENTS  AND  COTANGENTS. 

Tadle  IIL  1 

/ 

36° 

37° 

1 

38° 

3' 

)° 

/ 

Tangent.  C 

otang. 

Tangent.  C 

otang.  1 

Tangent.  C 

itang. 

Tangent. 

Cotang. 

0 

72654   I- 

37638 

75355   I. 

32704  1 

78129   1- 

27994 

80978 

1^23490 

60 

I 

72699   I. 
7274J   1- 

37554 

75401   !• 

32624 

78173   !■ 

27917 

81027 

I .23416 

?2 

2 

37470 

75447   I  • 

32  544 

78222   I- 

27841 

81075 

1^23343 

58 

3 

727S8   I- 

37386 

75492   I  • 

32164 

78269   1 . 

27764 

81123 

1.23270 

57 

4 

72832  I- 

37302 

75538  I. 

32384 

78316   !• 

27688 

81171 

I •23196 

56 

5 

72877  I. 

37218 

75584  I 

323o4 

78363  !• 

27611 

81220 

I^23l23 

55 

6 

72921   !• 

37134 

75629  I 

32224 

78410  1- 

27535 

81268 

i^23o5o 

54 

7 

72966   I • 

37050 

75673  I 

32144 

78457  1 

27458 

8i3i6 

1-22977 

53 

8 

73010   !• 

3696T 

73721  I 

32064 

78304  I 

27382 

8i364 

1 ■ 22904 

52 

9 

73o55  I- 

36883 

73767  I 

31984 

78551  I 

27306 

8i4i3 

1 -22831 

5i 

10 

73100   !• 

36800 

75812  I 

31904 

78593  I 

27230 

81461 

I -22758  i  5o  1 

II 

73i44  I- 

36716 

75858  I 

31825 

78645  I 

27153 

8i5io 

1 -2  2685 

ii 

12 

73i8g  I 

36633 

75904  I 

31745 

78692  I 

27077 

8i558 

I ■22612 

i3 

73234  I- 

36549 

75950  I 

3 1 665 

78739  I 

27001 

81606 

1 ■ 22539 

47 

i4 

73278  I- 

36466 

73996  I 

3 1 586 

78786  I 

26925 

8i655 

1-22467 

46 

i5 

73323  I 

36383 

76042  I 

3i5o7 

78834  1 

26849 

81703 

1-22394 

43 

i6 

73368  I 

363oo 

76088  I 

31427 

78881   I 

26774 

81752 

1-22321 

44 

17 

73413  I 

36217 

76134  1 

3 1 348 

78928  I 

26698 

81800 

1^22249 

43 

i8 

73437  I 

36i33 

76180  I 

3 1 269 

78975  I 

26622 

81849 

1^22176 

42 

'9 

73502  I 

36o5i 

76226  I 

31190 

79022  I 

26546 

81898 

1-22104 

41 

20 

73547  I 

35968 

76272  I 

3iiio 

79070  I 

26471 

81946 

I -2  2031 

40 

2! 

73592  I 

35885 

76318  1 

3io3i 

79117  I 

26395 

81995 

I ■21939 

l2 

22 

73637  I 

35802 

76364  I 

3oo52 

79164  1 

26319 

82044 

1 ■ 2 1 886 

38 

23 

73681  I 

35719 

76410  1 

30873 

79212  I 

26244 

82092 

I.21SI4 

37 

24 

73726  I 

35637 

76456  I 

30795 

79239  I 

26169 

82141 

1.21742 

36 

25 

73771  I 

35554 

765o2  I 

30716 

79306  I 

26093 

821QO 

1.21670 

35 

26 

73816  I 

35472 

76548  I 

3o637 

79354  I 

26018 

82238 

1.21598 

34 

S 

73861  I 

35389 

76594  I 

3o558 

79401  1 

25o43 

82287 

I^2l526 

33 

73906  I 

35307 

76640  I 

30480 

79449  1 

25867 

82336 

1 ^21454 

32 

29 

73951  I 

35224 

76686  I 

3  040 1 

79496  I 

25792 

82385 

I^2l382 

3i 

3o 

73996  I 

35i42 

76733  I 

3o323 

79344  1 

23717 

82434 

i^2i3io 

3o 

3i 

74041  I 

35o6o 

76779  ' 

3o244 

79591   I 

23642 

82483 

I^2I238 

29 

32 

74086  I 

3497« 

76825  1 

3oi66 

79639  I 

25367 

82531 

i^2ii66 

28 

33 

74i3i  I 

34896 

76871  I 

30087 

79686  I 

25492 

82580 

1^21094 

27 

34 

74176  I 

34814 

76918  1 

30009 

79734  I 

25417 

82629 

1^21023 

26 

35 

74221  I 

34732 

76964  1 

29931 

79781  I 

25343 

82678 

1^20951 

25 

36 

74267  I 

3465o 

77010  1 

29853 

79829  I 

23268 

82727 

1^20879 

24 

37 

74312  I 

34568 

77037  I 

29775 

79877  I 

25193 

82776 

1  ■  20808 

23 

38 

74357  I 

34487 

77103  I 

29696 

79924  1 

25ii8 

82825 

I  ■  20736 

22 

39 

74402  I 

34405 

77149  1 

29618 

79972  I 

23044 

82874 

I ■ 2o665 

21 

40 

74447  I 

34323 

77196  I 

29541 

80020  1 

24069 

82923 

1 -20593 

20 

41 

74492  I 

3^242 

77242  1 

29463 

80067  ' 

24895 

82972 

l-2o522 

'2 

42 

74538  I 

34160 

772S9  I 

29335 

801 1 5  1 

24820 

83022 

1-20451 

18 

43 

74583  I 

34079 

77335  . 

29307 

8oi63  I 

24746 

83071 

1-20379 

17 

44 

74628  I 

33998 

77382  I 

29229 

80211   1 

24672 

83120 

i-2o3o8 

16 

45 

74674  I 

33916 

77428  I 

29152 

80258  I 

24597 

83169 

1-20237 

i5 

46 

74719  I 

33835 

77475  1 

29074 

8o3o6  I 

24523 

832i8 

I -20166 

14 

47 

74764  I 

33754 

77321   I 

28997 

8o354  I 

24449 

83268 

1-20095 

i3 

48 

74810  I 

33673 

77568  I 

28919 

80402  I 

24373 

83317 

1-20024 

12 

49 

74855  I 

33592 

77613  1 

•28842 

80430  1 

243oi 

83366 

1-19933 

1 1 

5o 

74900  I 

335ii 

77661  I 

28764 

80498  I 

■24227 

8341 5 

I-I98S2 

10 

5i 

^4946  I 

33430 

777''8  i 

•2S687 

So546  I 

■24i53 

83465 

1-19811 

9 

52 

74991  I 

.33349 

77734  1 

•28610 

80594  I 

■24079 

835i4 

1-19740 

8 

53 

75037  I 

•33268 

77801  I 

■28533 

80642  I 

■  24003 

83564 

1 -19669 

7 

54 

75082  I 

•33187 

77848  I 

•28456 

80690  1 

■23931 

836 1 3 

I -19599 

6 

55 

75128  I 

■33107 

77S95  I 

•28379 

8073s  I 

■23858 

83662 

1-19328 

5 

56 

75173  I 

•33o26 

77941  1 

■2S3o2 

80786  I 

•23784 

83712 

1-19457 

4 

57 

75219  I 

•32946 

7^988  I 

■28225 

8o834  I 

•23710 

83761 

1-19387 

3 

58 

75264  I 

•32865 

78035  I 

•28148 

80882  I 

■23637 

838 11 

I -19316 

2 

59 

75310  I 

•32785 

78082   1 

■28071 

80930  I 

■23563 

83860 

1-19246 

I 

60 

75355  I 

•32704 

78129  I 

■27994 

80978  I 

•23490 

83910 

1-191751  0  1 

/ 

Cotang.  T 

angent 

Cotang.  j  T 

angeut. 

Cotang.  T 

angent. 

Cotang. 

Tangent. 

t 

1       roc 

1     5.3- 

5-2^ 

51° 

■ft 

5 

0° 

Table  III. 

NATURAL  TANGENTS  AND  COTANGEN'l 

S. 

83 

/ 

40° 

41° 

4 

2° 

43  = 

/ 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent. 

Cotang. 

Tangent.  C 

ui»ng. 

0 

83910 

1-19175 

86929 

i-i5o37 

90040 

1-I106l 

93252   I 

07237  60  1 

I 

83960 

1-19105 

86980 

1.14969 

90093 

I -10996 
1-10931 

93306   I 

07174 

59 

2 

84009 

I -19035 

87031 

1-14902 

90146 

93360   1 

07112 

58 

3 

840D9 

1. 1 8964 

87082 

r- 14834 

90199 

1-10867 

93415   I 

07049 

57 

4 

84108 

1-18894 

87133 

I -14767 

90231 

I -10802 

93469   1 

06987 

56 

5 

84 1 58 

1-18824 

87184 

1-14699 

9o3o4 

I. 10737 

93524   I 

06925 

55 

6 

84208 

1-18754 

87236 

1-146J2 

90357 

1-10672 

93578   I 

06862 

54 

7 

84258 

I -18684 

87281 
87338 

1-14565 

90410 

1-10607 

93633  I 

06800 

53 

8 

84307 

I -18614 

1-14498 

90463 

I-I0543 

93688  I 

06738 

52 

9 

84357 

1-18544 

87389 

1-14430 

9o5i6 

1-10478 

93742  I 

06676 

5i 

10 

84407 

1-18474 

87441 

1-14363 

90569 

1-10414 

93797  I 

066 1 3 

5o 

II 

84457 

I -18404 

87492 

1-14296 

90621 

I -10349 

93832  I 

06551 

40 
48 

12 

84507 

I -18334 

87543 

1-14229 

90674 

I -10285 

93906  I 

06489 

i3 

84556 

1-1S264 

87595 

i-i4i62 

90727 

I -10220 

93961  I 

06427 

47 

14 

84606 

1-18194 

87646 

1-14095 

90781 

i-ioi56 

94016  I 

06365 

46 

i5 

84656 

1-18125 

87698 

1-14028 

90834 

1-10091 

94071  I 

o63o3 

45 

i6 

84706 

i-i8o55 

87749 

I -13961 

90887 

1-10027 

94125  I 

06241 

44 

17 

84756 

1-17986 

87801 

1-13894 

90940 

1  -  09963 

94180  I 

06179 

43 

i8 

84806 

1-17916 

87852 

I -13828 

90993 

1-09899 

94235  I 

06117  1  42  1 

'9 

84356 

I -17846 

87904 

1-13761 

91046 

1-09834 

94290  1 

o6o56 

41 

20 

84906 

I-I7777 

87955 

I -13694 

91099 

1-09770 

94345  I 

03994 

40 

21 

84956 

1-17708 

88007 

I -13627 

91133 

1-09706 

9.1400  I 

05932 

39 

22 

85oo6 

I -17638 

88059 

i-i356i 

91206 

1-09642 

94455  I 

o5b70 

38 

23 

85o57 

I -17569 

88110 

I -13494 

91259 

1-09578 

94310  I 

03809 

P 

24 

85i07 

I -17500 

88 162 

I -13428 

9i3i3 

1-09514 

94565  I 

05747 

36 

2J 

85 157 

I -17430 

88214 

i-i336i 

91366 

I  -  09450 

94620  1 

o5685 

35 

26 

85207 

I -17361 

88265 

I -13295 

91419 
91473 

1-09386 

94676  1 

0.5624 

34 

11 

85257 

1-17292 

88317 

1-13228 

1-09322 

94731  1 

03562 

33 

85307 

1-17223 

88369 

i-i3i62 

91526 

1-09258 

94786  I 

o55oi 

32 

29 

85358 

1-17154 

88421 

I -13096 

9i58o 

1-09195 

94841  I 

05439 

3i 

3o 

85408 

I -17085 

88473 

i-i3o29 

91633 

1-09131 

94896  1 

05378 

3o 

3i 

85458 

1 -17016 

88524 

1-I2063 
1-12807 
I-I283I 

91687 

1  -  09067 

94952  1 

o53i7 

29 

32 

85  509 

1-16947 

88576 

91740 

I  - oooo3 

93007  I 

o5255 

28 

33 

85559 

1-16M 

88628 

9 '794 

1-08940 

95062  I 

o5i94 
o5i33 

27 

34 

85609 

1-16809 

88680 

I -12765 

91847 

1-08876 

93118  1 

26 

35 

85660 

1-16741 

88732 

\:\l^ 

91901 

I -08813 

93173  1 

o5o72 

25 

36 

85710 

1-16672 

88784 

91953 

1-08749 

93229  I 

o5oio 

24 

ll 

85761 

i-i66o3 

88836 

I -12567 

92008 

1-08686 

95284  I 

04949 

23 

858 II 

1-16535 

88888 

I-125oi 

92062 

1-08622 

95340  I 

04888 

22 

39 

85862 

1-16466 

88940 

I -12435 

92116 

1-08559 

95393  1 

04827 

21 

40 

85912 

1-16393: 

88992 

I • 1 2369 

92170 

I -08496 

95431  I 

04766 

20 

41 

85963 

1-16329! 

89045 

i-i23o3 

92223 

1-08432 

95506  I 

04705 

10 

18 

42 

86014 

1-16261 

89097 

I -12238 

92277 

1-08369 

95562  I 

04644 

43 

86064 

1-16192 

89149 

1-12472 

9233i 

I -08306 

95618  I 

04583 

17 

44 

861 1 5 

i-i6i24 

89201 

1-12106 

92385 

1.08243 

93673  I 

04522 

16 

45 

86166 

1 -16056 

89253 

I -12041 

92439 

1-08179 

95729  1 

04461 

i5 

.  46 

86216 

1-15987 

89306 

1-11975 

92493 

i-o8ii6 

95785  1 

04401 

14 

8 

86267 

1.15919 

89358 

1-11009 

92547 

i.o8o53 

93841  I 

04340 

i3 

863 18 

i-i585i 

89410 

1-11844 

92601 

1.07990 

95897  I 

04279 

12 

49 

86368 

I-I5783! 

89463 

1-11778 

92655 

1.07027 
1.07864 

90932  I 

04218 

II 

DO 

86419 

i-i57i5 ; 

89515 

1-11713 

92709 

96008  I 

04 1 58 

10 

5i 

86470 

1 • 1 5647  , 

89567 

1-11648 

92763 

1.07801 

96064  I 

04097 

t 

52 

86521 

1-15579 

89620 

I • 1 1 582 

92817 

1.07738 

96120  I 

040.36 

53 

86572 

i-i55ii 

89672 

i-ii5i7 

92872 

1.07676 

96176  I 

03976 

I 

54 

86623 

I -15443 

89725 

I -11452 

92926 

1-07613 

96232  I 

03915 

55 

86674 

I -15375 

89777 

1-11387 

92980 

1-07550 

96288  1 

o3855 

5 

56 

86725 

I -15308 

89830 

1-II321 

93o34 

1.07487 

96344  I 

03794 

4 

57 

86776 

i-i524o 

89883 

I -11256 

930S8 

1-07425 

96400  I 

03734 

3 

53 

86827 

i-i5i72 

89935 

I  11191 

93143 

1-07362 

96457  I 

03674 

a 

59 

86878 

im5io4 

8998S 

1  • 1 1 1 26 

93197 

1-07299 

965i3  1  I 

o36i3 

I. 

60 

86929 

i'i5o37 

90040 

1-11061 

93252 

1.07237 

96569  j  1 

o3553 

0' 

/ 

(  otang. 

Tangent. 

Cotang. 

Ta:)geut. 

Cotang. 

Tangent. 

Cotai.g.  1  T 

angent. 

/ 

4 

J  = 

4 

8  = 

4 

-0 

40  = 

l*-»  vv\^.^ 


-1 


t 


VV-Lb   <UA/t.«Il  /^V*««4— A/^^/iX^ 


ATURAL  TANGENTS  AND  COTANGENTS. 


Table  III. 


44° 


Cotan".' 


96369 
96625 
96681 
96738 
96794 
96830 
96907 
96963 
97020 
97076 
97 1 33 
97189 
97246 
97302 
97359 
97416 

97472 
97329 
97586 
97643 
97700 
97736 
97813 
97870 

97927 
97984 
9804 1 
98098 
98155 
98213 
98270 


Cotaiig. 


I  -03553 
1-03493 
i-o3433 
1-03372 
i-o33i2 

I -03232 

I -03 192 
I -03132 
1-03072 
i-o3oi2 

I -02032 

I -02892 

1-02832 

1-02772 
1-02713 
1-02653 

I -02593 
1-02533 

1-02474 
1-02414 
1-02355 
1 -02295 

1-02236 

1-02176 
1-02117 
I -02057 
1-01998 
I  01939 
1-01879 
I -01820 
I -0176 1 


Tangent. 


46= 


60 
5g 

58 

57 
56 
55 
54 
53 

52 

5i 
5o 

49 
48 
47 
46 
45 

44 
43 
42 
41 
40 

39 
38 

37 
36 
35 
34 
33 

32 

3i 
3o 


44c 


Tangent. 


Cotang 


3i 
32 

33 
34' 
35 
36 

37 
33 

39 
4o 
4i 

42 

43 
44 
45 

46 

47 
48 

49 
5o 
5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 


98327 
98384 
98441 

98499 
98556 
9S613 
98671 
98728 
98786 
98843 
98901 
98953 
99016 
99073 
99i3i 

99189 

99247 
99304 
99362 
99420 
99478 
99536 
99594 
99632 
99710 
99768 
99826 
99884 
99942 
Tnit. 


Cotang. 


01702 
01642 
01383 
oi524 
01465 
01406 
01 347 
01283 
01229 
01 170 
01112 
oio53 
00994 
00935 
00876 

00S18 
00739 
00701 
00642 
00383 
oo525 
00467 
00408 
oo35o 
00291 
00233 
00175 
001 16 
ooo53 
Unit. 


45= 


TABLE  OF  CONSTANTS. 

Base  of  Napier's  system  of  logarithms  =  t  =  2-718281828439 

Mod.  of  common  sj'st.  of  logarithms  =  ....  com.  log.  «  =  M  =  0-434294481903 

Eatio  of  circumference  to  diameter  of  a  circle  = jt  =  3 -14 1592653590 

log.  JT  =  0-497149872694 

TT-  =  9-S69604401089 y/  5r=  I -772453850906 

Arc  of  same  length  as  radius  = 180°  -j-  jr  =  10800'  -f-  w  =  648000"  -r-  r. 

180°  -H  ff  =  57° -2937795130, log.  =  1-758122632409 

10800'  4-  jr=  3437'-7467707849, log.  =  3-536273882793 

648000"  -f-5r=  206264"- 8062470964, log.  =  5 -3 14425 133176 

Tropical  year  =  365d.  5h.  48m.  47s.  -588  =  365d.  .242217456,  log.  =  2 -56258 10 
Sidereal  year  =  365d.  6h.  9m.  los.  -742  =  365d.  -256374332,  log.  =  2-5623978 
24h.  sol.  t.=24h.  3m.  56s.  '555335  sid.  t.=24h.Xi -00273791,  log.  1-002=0-0011874 
24h.sid.t.=24h.— (3m. 55s. '90944) sol.  t.=24h.Xo-9972696,  log.  0-997=9-9988126 

British  imperial  gallon  =  277 -  274  cubic  inches, log.  =  2-4429091 

Length  of  sec.  pend.,  in  inches,  at  London,  39-13929;   Paris,  39-1285;  New 

York,  39- 1285- 
French  metre  =  3 '2808992  Eiiglish/<3«i  =  39-3707904  inches. 
I  cubic  inch  of  water  (bar.  3o  inches,  Fahr.  therm.  62°)  =  252-458  Troy  grains. 


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